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How many $10$-digit numbers (allowing initial digit to be zero) in which only $5$ of the $10$ possible digits are represented?
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The answer I found was $$(5^10-|textonly~4~textdigits|-|textonly~ 3|-|textonly~ 2|-|textonly~1|) cdot C(10,5)=$$
where
$|textonly~1~textdigit| = 1^10 cdot C(10,1)$
$|textonly~2| = (2^10-|textonly~1|) cdot C(10,2)$
$|textonly~3| = (3^10-|textonly~2|-|textonly~1|) cdot C(10,3)$ ...etc
$5^10$ is supposed to represent the number of $10$ digit numbers which only use $5$ or fewer of $10$ distinct digits, just as there are $2^10$ binary strings of length $10$. I am subtracting from $5^10$ the number of strings which only contain $4$ distinct digits, $3$ distinct digits, $2$ distinct digits, and $1$ distinct digit, so I get the number of strings that use exactly $5$ distinct digits. I multiply by $C(10,5)$ because there are $C(10,5)$ ways to choose $5$ distinct digits. $|textonly~ 1|$ means the number of strings of length $10$ with exactly $1$ distinct digit.
For some reason I am not getting the right answer. Can someone tell me what I am doing wrong and post the correct answer please?
combinatorics permutations inclusion-exclusion
edited 16 hours ago
N. F. Taussig
37.9k92953
asked 2 days ago
Kevin
1
add a comment |Â
up vote
0
down vote
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The answer I found was $$(5^10-|textonly~4~textdigits|-|textonly~ 3|-|textonly~ 2|-|textonly~1|) cdot C(10,5)=$$
where
$|textonly~1~textdigit| = 1^10 cdot C(10,1)$
$|textonly~2| = (2^10-|textonly~1|) cdot C(10,2)$
$|textonly~3| = (3^10-|textonly~2|-|textonly~1|) cdot C(10,3)$ ...etc
$5^10$ is supposed to represent the number of $10$ digit numbers which only use $5$ or fewer of $10$ distinct digits, just as there are $2^10$ binary strings of length $10$. I am subtracting from $5^10$ the number of strings which only contain $4$ distinct digits, $3$ distinct digits, $2$ distinct digits, and $1$ distinct digit, so I get the number of strings that use exactly $5$ distinct digits. I multiply by $C(10,5)$ because there are $C(10,5)$ ways to choose $5$ distinct digits. $|textonly~ 1|$ means the number of strings of length $10$ with exactly $1$ distinct digit.
For some reason I am not getting the right answer. Can someone tell me what I am doing wrong and post the correct answer please?
combinatorics permutations inclusion-exclusion
edited 16 hours ago
N. F. Taussig
37.9k92953
asked 2 days ago
Kevin
1
3
Why do you multiply the whole thing by C(10,5)? What does the 5^10 represent? And why are you subtracting from 5^10? In sum: Can you please explain the thinking behind your formula to us? That way, we can probably pinpoint where you're going wrong. Right now, we can only make a somewhat educated guess as to what you were thinking when coming up with this formula. Also, providing you with the correct formula doesn't really help to correct your thinking either. If you want to improve your thinking on this, you'll have to tell us what your thinking is.
– Bram28
2 days ago
5^10 is supposed to represent the number of 10 digit numbers which only use 5 or fewer of 10 distinct digits, just as there are 2^10 binary strings of length 10. I am subtracting from 5^10 the number of strings which only contain 4 distinct digits, 3 distinct digits, 2 distinct digits, and 1 distinct digits so I get the number of strings that use exactly 5 distinct digits. I multiply by C(10,5) because there are C(10,5) ways to choose 5 distinct digits. |only 1| means the number of strings of length 10 with exactly 1 distinct digits
– Kevin
2 days ago
1
Do you mean only $5$ or exactly $5$ of the ten digits are represented in the string? You seem to be interested in calculating the number of strings in which exactly five digits appear.
– N. F. Taussig
2 days ago
Please read this MathJax tutorial, which explains how to typeset mathematics on this site.
– N. F. Taussig
16 hours ago
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The answer I found was $$(5^10-|textonly~4~textdigits|-|textonly~ 3|-|textonly~ 2|-|textonly~1|) cdot C(10,5)=$$
where
$|textonly~1~textdigit| = 1^10 cdot C(10,1)$
$|textonly~2| = (2^10-|textonly~1|) cdot C(10,2)$
$|textonly~3| = (3^10-|textonly~2|-|textonly~1|) cdot C(10,3)$ ...etc
$5^10$ is supposed to represent the number of $10$ digit numbers which only use $5$ or fewer of $10$ distinct digits, just as there are $2^10$ binary strings of length $10$. I am subtracting from $5^10$ the number of strings which only contain $4$ distinct digits, $3$ distinct digits, $2$ distinct digits, and $1$ distinct digit, so I get the number of strings that use exactly $5$ distinct digits. I multiply by $C(10,5)$ because there are $C(10,5)$ ways to choose $5$ distinct digits. $|textonly~ 1|$ means the number of strings of length $10$ with exactly $1$ distinct digit.
For some reason I am not getting the right answer. Can someone tell me what I am doing wrong and post the correct answer please?
combinatorics permutations inclusion-exclusion
edited 16 hours ago
N. F. Taussig
37.9k92953
asked 2 days ago
Kevin
1
The answer I found was $$(5^10-|textonly~4~textdigits|-|textonly~ 3|-|textonly~ 2|-|textonly~1|) cdot C(10,5)=$$
where
$|textonly~1~textdigit| = 1^10 cdot C(10,1)$
$|textonly~2| = (2^10-|textonly~1|) cdot C(10,2)$
$|textonly~3| = (3^10-|textonly~2|-|textonly~1|) cdot C(10,3)$ ...etc
$5^10$ is supposed to represent the number of $10$ digit numbers which only use $5$ or fewer of $10$ distinct digits, just as there are $2^10$ binary strings of length $10$. I am subtracting from $5^10$ the number of strings which only contain $4$ distinct digits, $3$ distinct digits, $2$ distinct digits, and $1$ distinct digit, so I get the number of strings that use exactly $5$ distinct digits. I multiply by $C(10,5)$ because there are $C(10,5)$ ways to choose $5$ distinct digits. $|textonly~ 1|$ means the number of strings of length $10$ with exactly $1$ distinct digit.
For some reason I am not getting the right answer. Can someone tell me what I am doing wrong and post the correct answer please?
combinatorics permutations inclusion-exclusion
edited 16 hours ago
N. F. Taussig
37.9k92953
asked 2 days ago
Kevin
1
edited 16 hours ago
N. F. Taussig
37.9k92953
edited 16 hours ago
N. F. Taussig
37.9k92953
edited 16 hours ago
N. F. Taussig
37.9k92953
37.9k92953
asked 2 days ago
Kevin
1
asked 2 days ago
Kevin
1
asked 2 days ago
Kevin
1
1
3
Why do you multiply the whole thing by C(10,5)? What does the 5^10 represent? And why are you subtracting from 5^10? In sum: Can you please explain the thinking behind your formula to us? That way, we can probably pinpoint where you're going wrong. Right now, we can only make a somewhat educated guess as to what you were thinking when coming up with this formula. Also, providing you with the correct formula doesn't really help to correct your thinking either. If you want to improve your thinking on this, you'll have to tell us what your thinking is.
– Bram28
2 days ago
5^10 is supposed to represent the number of 10 digit numbers which only use 5 or fewer of 10 distinct digits, just as there are 2^10 binary strings of length 10. I am subtracting from 5^10 the number of strings which only contain 4 distinct digits, 3 distinct digits, 2 distinct digits, and 1 distinct digits so I get the number of strings that use exactly 5 distinct digits. I multiply by C(10,5) because there are C(10,5) ways to choose 5 distinct digits. |only 1| means the number of strings of length 10 with exactly 1 distinct digits
– Kevin
2 days ago
1
Do you mean only $5$ or exactly $5$ of the ten digits are represented in the string? You seem to be interested in calculating the number of strings in which exactly five digits appear.
– N. F. Taussig
2 days ago
Please read this MathJax tutorial, which explains how to typeset mathematics on this site.
– N. F. Taussig
16 hours ago
add a comment |Â
3
Why do you multiply the whole thing by C(10,5)? What does the 5^10 represent? And why are you subtracting from 5^10? In sum: Can you please explain the thinking behind your formula to us? That way, we can probably pinpoint where you're going wrong. Right now, we can only make a somewhat educated guess as to what you were thinking when coming up with this formula. Also, providing you with the correct formula doesn't really help to correct your thinking either. If you want to improve your thinking on this, you'll have to tell us what your thinking is.
– Bram28
2 days ago
5^10 is supposed to represent the number of 10 digit numbers which only use 5 or fewer of 10 distinct digits, just as there are 2^10 binary strings of length 10. I am subtracting from 5^10 the number of strings which only contain 4 distinct digits, 3 distinct digits, 2 distinct digits, and 1 distinct digits so I get the number of strings that use exactly 5 distinct digits. I multiply by C(10,5) because there are C(10,5) ways to choose 5 distinct digits. |only 1| means the number of strings of length 10 with exactly 1 distinct digits
– Kevin
2 days ago
1
Do you mean only $5$ or exactly $5$ of the ten digits are represented in the string? You seem to be interested in calculating the number of strings in which exactly five digits appear.
– N. F. Taussig
2 days ago
Please read this MathJax tutorial, which explains how to typeset mathematics on this site.
– N. F. Taussig
16 hours ago
3
3
Why do you multiply the whole thing by C(10,5)? What does the 5^10 represent? And why are you subtracting from 5^10? In sum: Can you please explain the thinking behind your formula to us? That way, we can probably pinpoint where you're going wrong. Right now, we can only make a somewhat educated guess as to what you were thinking when coming up with this formula. Also, providing you with the correct formula doesn't really help to correct your thinking either. If you want to improve your thinking on this, you'll have to tell us what your thinking is.
– Bram28
2 days ago
Why do you multiply the whole thing by C(10,5)? What does the 5^10 represent? And why are you subtracting from 5^10? In sum: Can you please explain the thinking behind your formula to us? That way, we can probably pinpoint where you're going wrong. Right now, we can only make a somewhat educated guess as to what you were thinking when coming up with this formula. Also, providing you with the correct formula doesn't really help to correct your thinking either. If you want to improve your thinking on this, you'll have to tell us what your thinking is.
– Bram28
2 days ago
5^10 is supposed to represent the number of 10 digit numbers which only use 5 or fewer of 10 distinct digits, just as there are 2^10 binary strings of length 10. I am subtracting from 5^10 the number of strings which only contain 4 distinct digits, 3 distinct digits, 2 distinct digits, and 1 distinct digits so I get the number of strings that use exactly 5 distinct digits. I multiply by C(10,5) because there are C(10,5) ways to choose 5 distinct digits. |only 1| means the number of strings of length 10 with exactly 1 distinct digits
– Kevin
2 days ago
5^10 is supposed to represent the number of 10 digit numbers which only use 5 or fewer of 10 distinct digits, just as there are 2^10 binary strings of length 10. I am subtracting from 5^10 the number of strings which only contain 4 distinct digits, 3 distinct digits, 2 distinct digits, and 1 distinct digits so I get the number of strings that use exactly 5 distinct digits. I multiply by C(10,5) because there are C(10,5) ways to choose 5 distinct digits. |only 1| means the number of strings of length 10 with exactly 1 distinct digits
– Kevin
2 days ago
1
1
Do you mean only $5$ or exactly $5$ of the ten digits are represented in the string? You seem to be interested in calculating the number of strings in which exactly five digits appear.
– N. F. Taussig
2 days ago
Do you mean only $5$ or exactly $5$ of the ten digits are represented in the string? You seem to be interested in calculating the number of strings in which exactly five digits appear.
– N. F. Taussig
2 days ago
Please read this MathJax tutorial, which explains how to typeset mathematics on this site.
– N. F. Taussig
16 hours ago
Please read this MathJax tutorial, which explains how to typeset mathematics on this site.
– N. F. Taussig
16 hours ago
add a comment |Â
1 Answer
1
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0
down vote
Your $C(10,5)$ chooses which five digits will be used. When you want to select the strings that use only one digit, you only want to subtract the ones that use one of these five digits, so it should be $C(5,1)$
answered 2 days ago
Ross Millikan
275k21183348
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Your $C(10,5)$ chooses which five digits will be used. When you want to select the strings that use only one digit, you only want to subtract the ones that use one of these five digits, so it should be $C(5,1)$
answered 2 days ago
Ross Millikan
275k21183348
add a comment |Â
up vote
0
down vote
Your $C(10,5)$ chooses which five digits will be used. When you want to select the strings that use only one digit, you only want to subtract the ones that use one of these five digits, so it should be $C(5,1)$
answered 2 days ago
Ross Millikan
275k21183348
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Your $C(10,5)$ chooses which five digits will be used. When you want to select the strings that use only one digit, you only want to subtract the ones that use one of these five digits, so it should be $C(5,1)$
answered 2 days ago
Ross Millikan
275k21183348
Your $C(10,5)$ chooses which five digits will be used. When you want to select the strings that use only one digit, you only want to subtract the ones that use one of these five digits, so it should be $C(5,1)$
answered 2 days ago
Ross Millikan
275k21183348
answered 2 days ago
Ross Millikan
275k21183348
answered 2 days ago
Ross Millikan
275k21183348
answered 2 days ago
Ross Millikan
275k21183348
275k21183348
add a comment |Â
add a comment |Â
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3
Why do you multiply the whole thing by C(10,5)? What does the 5^10 represent? And why are you subtracting from 5^10? In sum: Can you please explain the thinking behind your formula to us? That way, we can probably pinpoint where you're going wrong. Right now, we can only make a somewhat educated guess as to what you were thinking when coming up with this formula. Also, providing you with the correct formula doesn't really help to correct your thinking either. If you want to improve your thinking on this, you'll have to tell us what your thinking is.
– Bram28
2 days ago
5^10 is supposed to represent the number of 10 digit numbers which only use 5 or fewer of 10 distinct digits, just as there are 2^10 binary strings of length 10. I am subtracting from 5^10 the number of strings which only contain 4 distinct digits, 3 distinct digits, 2 distinct digits, and 1 distinct digits so I get the number of strings that use exactly 5 distinct digits. I multiply by C(10,5) because there are C(10,5) ways to choose 5 distinct digits. |only 1| means the number of strings of length 10 with exactly 1 distinct digits
– Kevin
2 days ago
1
Do you mean only $5$ or exactly $5$ of the ten digits are represented in the string? You seem to be interested in calculating the number of strings in which exactly five digits appear.
– N. F. Taussig
2 days ago
Please read this MathJax tutorial, which explains how to typeset mathematics on this site.
– N. F. Taussig
16 hours ago