Mixing
Chessboard possible ways from A to B [duplicate]
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1
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This question already has an answer here:
chess path combination
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How can I calculate the possible ways from point A to point B on a empty 8x8 chessboard while only being allowed to move up and right.
Lets say I have to travel from
x1 = 1, y1 = 1
to
x2 = 2, y2 = 2
Then there are 2 possible ways.
But what about
x1 = 1, y1 = 1
to
x2 = 8, y2 = 8
Is there some sort of a formula? Something with vectors?
combinatorics chessboard
edited Aug 4 at 6:43
Jyrki Lahtonen
104k12161355
asked Aug 3 at 12:11
Gaterde
1085
marked as duplicate by Jyrki Lahtonen, amWhy, José Carlos Santos, max_zorn, Adrian Keister 2 days ago
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
 |Â
show 1 more comment
up vote
1
down vote
favorite
This question already has an answer here:
chess path combination
1 answer
How can I calculate the possible ways from point A to point B on a empty 8x8 chessboard while only being allowed to move up and right.
Lets say I have to travel from
x1 = 1, y1 = 1
to
x2 = 2, y2 = 2
Then there are 2 possible ways.
But what about
x1 = 1, y1 = 1
to
x2 = 8, y2 = 8
Is there some sort of a formula? Something with vectors?
combinatorics chessboard
edited Aug 4 at 6:43
Jyrki Lahtonen
104k12161355
asked Aug 3 at 12:11
Gaterde
1085
marked as duplicate by Jyrki Lahtonen, amWhy, José Carlos Santos, max_zorn, Adrian Keister 2 days ago
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
5
Hint: You have to move as many times up as the difference of $y$'s is; and you have to move right as many times as the difference if $x$'s is. All the different ways of getting from the starting point to the ending point are just reorderings of these up/right moves.
– Matti P.
Aug 3 at 12:14
1
$(8+8)!/(8!8!)$
– daruma
Aug 3 at 12:17
@daruma If you add eight to $x_1=1$ you get $x_2=x_1+8=9$.
– Jyrki Lahtonen
Aug 4 at 6:42
Somewhat related.
– Jyrki Lahtonen
Aug 4 at 7:16
And a duplicate of this
– Jyrki Lahtonen
Aug 4 at 7:17
 |Â
show 1 more comment
up vote
1
down vote
favorite
up vote
1
down vote
favorite
This question already has an answer here:
chess path combination
1 answer
How can I calculate the possible ways from point A to point B on a empty 8x8 chessboard while only being allowed to move up and right.
Lets say I have to travel from
x1 = 1, y1 = 1
to
x2 = 2, y2 = 2
Then there are 2 possible ways.
But what about
x1 = 1, y1 = 1
to
x2 = 8, y2 = 8
Is there some sort of a formula? Something with vectors?
combinatorics chessboard
edited Aug 4 at 6:43
Jyrki Lahtonen
104k12161355
asked Aug 3 at 12:11
Gaterde
1085
This question already has an answer here:
chess path combination
1 answer
How can I calculate the possible ways from point A to point B on a empty 8x8 chessboard while only being allowed to move up and right.
Lets say I have to travel from
x1 = 1, y1 = 1
to
x2 = 2, y2 = 2
Then there are 2 possible ways.
But what about
x1 = 1, y1 = 1
to
x2 = 8, y2 = 8
Is there some sort of a formula? Something with vectors?
This question already has an answer here:
chess path combination
1 answer
combinatorics chessboard
edited Aug 4 at 6:43
Jyrki Lahtonen
104k12161355
asked Aug 3 at 12:11
Gaterde
1085
edited Aug 4 at 6:43
Jyrki Lahtonen
104k12161355
edited Aug 4 at 6:43
Jyrki Lahtonen
104k12161355
edited Aug 4 at 6:43
Jyrki Lahtonen
104k12161355
104k12161355
asked Aug 3 at 12:11
Gaterde
1085
asked Aug 3 at 12:11
Gaterde
1085
asked Aug 3 at 12:11
Gaterde
1085
1085
marked as duplicate by Jyrki Lahtonen, amWhy, José Carlos Santos, max_zorn, Adrian Keister 2 days ago
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Jyrki Lahtonen, amWhy, José Carlos Santos, max_zorn, Adrian Keister 2 days ago
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
5
Hint: You have to move as many times up as the difference of $y$'s is; and you have to move right as many times as the difference if $x$'s is. All the different ways of getting from the starting point to the ending point are just reorderings of these up/right moves.
– Matti P.
Aug 3 at 12:14
1
$(8+8)!/(8!8!)$
– daruma
Aug 3 at 12:17
@daruma If you add eight to $x_1=1$ you get $x_2=x_1+8=9$.
– Jyrki Lahtonen
Aug 4 at 6:42
Somewhat related.
– Jyrki Lahtonen
Aug 4 at 7:16
And a duplicate of this
– Jyrki Lahtonen
Aug 4 at 7:17
 |Â
show 1 more comment
5
Hint: You have to move as many times up as the difference of $y$'s is; and you have to move right as many times as the difference if $x$'s is. All the different ways of getting from the starting point to the ending point are just reorderings of these up/right moves.
– Matti P.
Aug 3 at 12:14
1
$(8+8)!/(8!8!)$
– daruma
Aug 3 at 12:17
@daruma If you add eight to $x_1=1$ you get $x_2=x_1+8=9$.
– Jyrki Lahtonen
Aug 4 at 6:42
Somewhat related.
– Jyrki Lahtonen
Aug 4 at 7:16
And a duplicate of this
– Jyrki Lahtonen
Aug 4 at 7:17
5
5
Hint: You have to move as many times up as the difference of $y$'s is; and you have to move right as many times as the difference if $x$'s is. All the different ways of getting from the starting point to the ending point are just reorderings of these up/right moves.
– Matti P.
Aug 3 at 12:14
Hint: You have to move as many times up as the difference of $y$'s is; and you have to move right as many times as the difference if $x$'s is. All the different ways of getting from the starting point to the ending point are just reorderings of these up/right moves.
– Matti P.
Aug 3 at 12:14
1
1
$(8+8)!/(8!8!)$
– daruma
Aug 3 at 12:17
$(8+8)!/(8!8!)$
– daruma
Aug 3 at 12:17
@daruma If you add eight to $x_1=1$ you get $x_2=x_1+8=9$.
– Jyrki Lahtonen
Aug 4 at 6:42
@daruma If you add eight to $x_1=1$ you get $x_2=x_1+8=9$.
– Jyrki Lahtonen
Aug 4 at 6:42
Somewhat related.
– Jyrki Lahtonen
Aug 4 at 7:16
Somewhat related.
– Jyrki Lahtonen
Aug 4 at 7:16
And a duplicate of this
– Jyrki Lahtonen
Aug 4 at 7:17
And a duplicate of this
– Jyrki Lahtonen
Aug 4 at 7:17
 |Â
show 1 more comment
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
We need to make in any case
- $8$ step on the right
- $8$ step up
therefore the number of possible ways is
$$binom168=frac16!(8!)^2=frac16cdot 15cdot 14 cdot 13 cdot 12 cdot 11 cdot 10cdot 98cdot 7cdot 6 cdot 5 cdot 4 cdot 3 cdot2cdot 1=12,870$$
since once we have chosen the 8 step on the right (or up) the path is completely determined.
answered Aug 3 at 12:18
gimusi
63.7k73480
1
Could it be 7 and 14 instead, what with the player already starting in the corner A1? Right method, in any case.
– Chessanator
Aug 3 at 12:46
@Chessanator In that case we have $binom217=binom2114$. What is the corner A1?
– gimusi
Aug 3 at 12:48
Sorry: I was using chess notation for the bottom left corner. If the player already starts there after zero moves, then there's only seven steps right and seven up, for an answer of $binom147$. Ambiguity in the question results in a classic off by one error.
– Chessanator
Aug 3 at 12:51
@Chessanator Yes of course!
– gimusi
Aug 3 at 12:53
What @Chessanator says. On an 8x8 chessboard It is only 7 steps from the leftmost file to the rightmost file (denoted A and H respectively when using algebraic notation). Similarly it is only 7 steps from the 1st row to the eighth.
– Jyrki Lahtonen
Aug 4 at 6:41
 |Â
show 1 more comment
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
We need to make in any case
- $8$ step on the right
- $8$ step up
therefore the number of possible ways is
$$binom168=frac16!(8!)^2=frac16cdot 15cdot 14 cdot 13 cdot 12 cdot 11 cdot 10cdot 98cdot 7cdot 6 cdot 5 cdot 4 cdot 3 cdot2cdot 1=12,870$$
since once we have chosen the 8 step on the right (or up) the path is completely determined.
answered Aug 3 at 12:18
gimusi
63.7k73480
1
Could it be 7 and 14 instead, what with the player already starting in the corner A1? Right method, in any case.
– Chessanator
Aug 3 at 12:46
@Chessanator In that case we have $binom217=binom2114$. What is the corner A1?
– gimusi
Aug 3 at 12:48
Sorry: I was using chess notation for the bottom left corner. If the player already starts there after zero moves, then there's only seven steps right and seven up, for an answer of $binom147$. Ambiguity in the question results in a classic off by one error.
– Chessanator
Aug 3 at 12:51
@Chessanator Yes of course!
– gimusi
Aug 3 at 12:53
What @Chessanator says. On an 8x8 chessboard It is only 7 steps from the leftmost file to the rightmost file (denoted A and H respectively when using algebraic notation). Similarly it is only 7 steps from the 1st row to the eighth.
– Jyrki Lahtonen
Aug 4 at 6:41
 |Â
show 1 more comment
up vote
1
down vote
accepted
We need to make in any case
- $8$ step on the right
- $8$ step up
therefore the number of possible ways is
$$binom168=frac16!(8!)^2=frac16cdot 15cdot 14 cdot 13 cdot 12 cdot 11 cdot 10cdot 98cdot 7cdot 6 cdot 5 cdot 4 cdot 3 cdot2cdot 1=12,870$$
since once we have chosen the 8 step on the right (or up) the path is completely determined.
answered Aug 3 at 12:18
gimusi
63.7k73480
1
Could it be 7 and 14 instead, what with the player already starting in the corner A1? Right method, in any case.
– Chessanator
Aug 3 at 12:46
@Chessanator In that case we have $binom217=binom2114$. What is the corner A1?
– gimusi
Aug 3 at 12:48
Sorry: I was using chess notation for the bottom left corner. If the player already starts there after zero moves, then there's only seven steps right and seven up, for an answer of $binom147$. Ambiguity in the question results in a classic off by one error.
– Chessanator
Aug 3 at 12:51
@Chessanator Yes of course!
– gimusi
Aug 3 at 12:53
What @Chessanator says. On an 8x8 chessboard It is only 7 steps from the leftmost file to the rightmost file (denoted A and H respectively when using algebraic notation). Similarly it is only 7 steps from the 1st row to the eighth.
– Jyrki Lahtonen
Aug 4 at 6:41
 |Â
show 1 more comment
up vote
1
down vote
accepted
up vote
1
down vote
accepted
We need to make in any case
- $8$ step on the right
- $8$ step up
therefore the number of possible ways is
$$binom168=frac16!(8!)^2=frac16cdot 15cdot 14 cdot 13 cdot 12 cdot 11 cdot 10cdot 98cdot 7cdot 6 cdot 5 cdot 4 cdot 3 cdot2cdot 1=12,870$$
since once we have chosen the 8 step on the right (or up) the path is completely determined.
answered Aug 3 at 12:18
gimusi
63.7k73480
We need to make in any case
- $8$ step on the right
- $8$ step up
therefore the number of possible ways is
$$binom168=frac16!(8!)^2=frac16cdot 15cdot 14 cdot 13 cdot 12 cdot 11 cdot 10cdot 98cdot 7cdot 6 cdot 5 cdot 4 cdot 3 cdot2cdot 1=12,870$$
since once we have chosen the 8 step on the right (or up) the path is completely determined.
answered Aug 3 at 12:18
gimusi
63.7k73480
answered Aug 3 at 12:18
gimusi
63.7k73480
answered Aug 3 at 12:18
gimusi
63.7k73480
answered Aug 3 at 12:18
gimusi
63.7k73480
63.7k73480
1
Could it be 7 and 14 instead, what with the player already starting in the corner A1? Right method, in any case.
– Chessanator
Aug 3 at 12:46
@Chessanator In that case we have $binom217=binom2114$. What is the corner A1?
– gimusi
Aug 3 at 12:48
Sorry: I was using chess notation for the bottom left corner. If the player already starts there after zero moves, then there's only seven steps right and seven up, for an answer of $binom147$. Ambiguity in the question results in a classic off by one error.
– Chessanator
Aug 3 at 12:51
@Chessanator Yes of course!
– gimusi
Aug 3 at 12:53
What @Chessanator says. On an 8x8 chessboard It is only 7 steps from the leftmost file to the rightmost file (denoted A and H respectively when using algebraic notation). Similarly it is only 7 steps from the 1st row to the eighth.
– Jyrki Lahtonen
Aug 4 at 6:41
 |Â
show 1 more comment
1
Could it be 7 and 14 instead, what with the player already starting in the corner A1? Right method, in any case.
– Chessanator
Aug 3 at 12:46
@Chessanator In that case we have $binom217=binom2114$. What is the corner A1?
– gimusi
Aug 3 at 12:48
Sorry: I was using chess notation for the bottom left corner. If the player already starts there after zero moves, then there's only seven steps right and seven up, for an answer of $binom147$. Ambiguity in the question results in a classic off by one error.
– Chessanator
Aug 3 at 12:51
@Chessanator Yes of course!
– gimusi
Aug 3 at 12:53
What @Chessanator says. On an 8x8 chessboard It is only 7 steps from the leftmost file to the rightmost file (denoted A and H respectively when using algebraic notation). Similarly it is only 7 steps from the 1st row to the eighth.
– Jyrki Lahtonen
Aug 4 at 6:41
1
1
Could it be 7 and 14 instead, what with the player already starting in the corner A1? Right method, in any case.
– Chessanator
Aug 3 at 12:46
Could it be 7 and 14 instead, what with the player already starting in the corner A1? Right method, in any case.
– Chessanator
Aug 3 at 12:46
@Chessanator In that case we have $binom217=binom2114$. What is the corner A1?
– gimusi
Aug 3 at 12:48
@Chessanator In that case we have $binom217=binom2114$. What is the corner A1?
– gimusi
Aug 3 at 12:48
Sorry: I was using chess notation for the bottom left corner. If the player already starts there after zero moves, then there's only seven steps right and seven up, for an answer of $binom147$. Ambiguity in the question results in a classic off by one error.
– Chessanator
Aug 3 at 12:51
Sorry: I was using chess notation for the bottom left corner. If the player already starts there after zero moves, then there's only seven steps right and seven up, for an answer of $binom147$. Ambiguity in the question results in a classic off by one error.
– Chessanator
Aug 3 at 12:51
@Chessanator Yes of course!
– gimusi
Aug 3 at 12:53
@Chessanator Yes of course!
– gimusi
Aug 3 at 12:53
What @Chessanator says. On an 8x8 chessboard It is only 7 steps from the leftmost file to the rightmost file (denoted A and H respectively when using algebraic notation). Similarly it is only 7 steps from the 1st row to the eighth.
– Jyrki Lahtonen
Aug 4 at 6:41
What @Chessanator says. On an 8x8 chessboard It is only 7 steps from the leftmost file to the rightmost file (denoted A and H respectively when using algebraic notation). Similarly it is only 7 steps from the 1st row to the eighth.
– Jyrki Lahtonen
Aug 4 at 6:41
 |Â
show 1 more comment
5
Hint: You have to move as many times up as the difference of $y$'s is; and you have to move right as many times as the difference if $x$'s is. All the different ways of getting from the starting point to the ending point are just reorderings of these up/right moves.
– Matti P.
Aug 3 at 12:14
1
$(8+8)!/(8!8!)$
– daruma
Aug 3 at 12:17
@daruma If you add eight to $x_1=1$ you get $x_2=x_1+8=9$.
– Jyrki Lahtonen
Aug 4 at 6:42
Somewhat related.
– Jyrki Lahtonen
Aug 4 at 7:16
And a duplicate of this
– Jyrki Lahtonen
Aug 4 at 7:17