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Arranging cats and dogs - what is wrong with my approach
The Next CEO of Stack OverflowFour dogs and five cats race. In how many ways can this occur?What's wrong with my permutation logic?Find the number of ways this can be arranged in which no 2 women and no 2 men sit together given 4 men and 3 women are seated in a dinner table?In how many ways can the letters of word $PERMUTATIONS$ be arranged if there are always 4 letters between P and S?What is wrong in my attempt in permutations?Story Of naive Cats And Machiavellian MonkeyArranging $A$'s and $B$'s.Permutations of finishing a raceHow many strings of $6$ digits are there which use only the digits $0, 1$, or $2$ and in which $2$, whenever it appears, always does so after $1$?In how many ways can $n$ dogs and $k$ cats be arranged in a row so that no two cats are adjacent?
$begingroup$
We have 4 dogs and 3 cats in a line but no two cats can be together, in how many ways can they be arranged?
Since there are 5 spaces the cats can be in with the dogs fixed, there are $5 choose 3 * 4! * 3! = 1440$ ways and this is the correct answer.
I thought of a different approach. Instead of fixing the dogs' places, I fixed the places of the cats. Now, we have 4 spaces of which the two spaces in the middle must be filled. Therefore, out of the 4 dogs, 2 must fill those, and there are $4 * 3$ ways of doing this (since one dog must be chosen to fill one middle space and the other, to fill the second but now there are only 3 dogs left.)
The other two dogs are free to go to any of the 4 spaces, with $4^2$ possibilities.
The cats can now be arranged in $3!$ ways.
So, our final answer should be $3! * 4^2 * 4 * 3 = 1152$
Where have I gone wrong?
combinatorics permutations
asked yesterday
Akshat AgarwalAkshat Agarwal
512
New contributor
Akshat Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
We have 4 dogs and 3 cats in a line but no two cats can be together, in how many ways can they be arranged?
Since there are 5 spaces the cats can be in with the dogs fixed, there are $5 choose 3 * 4! * 3! = 1440$ ways and this is the correct answer.
I thought of a different approach. Instead of fixing the dogs' places, I fixed the places of the cats. Now, we have 4 spaces of which the two spaces in the middle must be filled. Therefore, out of the 4 dogs, 2 must fill those, and there are $4 * 3$ ways of doing this (since one dog must be chosen to fill one middle space and the other, to fill the second but now there are only 3 dogs left.)
The other two dogs are free to go to any of the 4 spaces, with $4^2$ possibilities.
The cats can now be arranged in $3!$ ways.
So, our final answer should be $3! * 4^2 * 4 * 3 = 1152$
Where have I gone wrong?
combinatorics permutations
asked yesterday
Akshat AgarwalAkshat Agarwal
512
New contributor
Akshat Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
2
$begingroup$
Is your problem arising as a consequence of a rainfall ?
$endgroup$
– Jean Marie
yesterday
add a comment |
$begingroup$
We have 4 dogs and 3 cats in a line but no two cats can be together, in how many ways can they be arranged?
Since there are 5 spaces the cats can be in with the dogs fixed, there are $5 choose 3 * 4! * 3! = 1440$ ways and this is the correct answer.
I thought of a different approach. Instead of fixing the dogs' places, I fixed the places of the cats. Now, we have 4 spaces of which the two spaces in the middle must be filled. Therefore, out of the 4 dogs, 2 must fill those, and there are $4 * 3$ ways of doing this (since one dog must be chosen to fill one middle space and the other, to fill the second but now there are only 3 dogs left.)
The other two dogs are free to go to any of the 4 spaces, with $4^2$ possibilities.
The cats can now be arranged in $3!$ ways.
So, our final answer should be $3! * 4^2 * 4 * 3 = 1152$
Where have I gone wrong?
combinatorics permutations
asked yesterday
Akshat AgarwalAkshat Agarwal
512
New contributor
Akshat Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
We have 4 dogs and 3 cats in a line but no two cats can be together, in how many ways can they be arranged?
Since there are 5 spaces the cats can be in with the dogs fixed, there are $5 choose 3 * 4! * 3! = 1440$ ways and this is the correct answer.
I thought of a different approach. Instead of fixing the dogs' places, I fixed the places of the cats. Now, we have 4 spaces of which the two spaces in the middle must be filled. Therefore, out of the 4 dogs, 2 must fill those, and there are $4 * 3$ ways of doing this (since one dog must be chosen to fill one middle space and the other, to fill the second but now there are only 3 dogs left.)
The other two dogs are free to go to any of the 4 spaces, with $4^2$ possibilities.
The cats can now be arranged in $3!$ ways.
So, our final answer should be $3! * 4^2 * 4 * 3 = 1152$
Where have I gone wrong?
combinatorics permutations
combinatorics permutations
asked yesterday
Akshat AgarwalAkshat Agarwal
512
New contributor
Akshat Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
Akshat AgarwalAkshat Agarwal
512
New contributor
Akshat Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
Akshat AgarwalAkshat Agarwal
512
New contributor
Akshat Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
Akshat AgarwalAkshat Agarwal
512
asked yesterday
Akshat AgarwalAkshat Agarwal
512
512
New contributor
Akshat Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Akshat Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Akshat Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
$begingroup$
Is your problem arising as a consequence of a rainfall ?
$endgroup$
– Jean Marie
yesterday
add a comment |
2
$begingroup$
Is your problem arising as a consequence of a rainfall ?
$endgroup$
– Jean Marie
yesterday
2
2
$begingroup$
Is your problem arising as a consequence of a rainfall ?
$endgroup$
– Jean Marie
yesterday
$begingroup$
Is your problem arising as a consequence of a rainfall ?
$endgroup$
– Jean Marie
yesterday
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The second computation is missing a symmetry. Say your initial pattern is $$underline quadC_1underline quadC_2underline quadC_3underline quad$$
You then populate the spaces immediately to the right of $C_1$, and $C_2$. As:
$$underline quadC_1D_1underline quadC_2D_2underline quadC_3underline quad$$
So far so good. You still have $D_3,D_4$ to place. Where can they go? True, they can each go to any of the four spaces, but if, say, they both go to the first space, in which order do they go?
Taking the two possible orders into account, we see that you are missing $$4times 3!times 4times 3=288$$ cases. Adding them back gives you the desired result.
Phrased differently: once you have placed $D_3$ there are now five available spaces for $D_4$ (since $D_4$ might go either to the left or to the right of $D_3$). thus you should have had $$3!times 4times 5times 4times 3=1440$$
answered yesterday
lulululu
43.3k25080
$endgroup$
add a comment |
$begingroup$
You can separate the two cases for the two last dogs: single dogs and double dogs.
Single dogs:
$$P(4,2)=frac4!2!=12.$$
Double dogs:
$$P(2,2)cdot C(4,1)=2cdot 4=8.$$
Hence, there are $12+8=20$ (not $4^2=16$) ways to distribute the last two dogs.
The final answer is:
$$3!cdot 20cdot 4cdot 3=1440.$$
answered yesterday
farruhotafarruhota
21.8k2842
$endgroup$
add a comment |
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2 Answers
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2 Answers
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votes
$begingroup$
The second computation is missing a symmetry. Say your initial pattern is $$underline quadC_1underline quadC_2underline quadC_3underline quad$$
You then populate the spaces immediately to the right of $C_1$, and $C_2$. As:
$$underline quadC_1D_1underline quadC_2D_2underline quadC_3underline quad$$
So far so good. You still have $D_3,D_4$ to place. Where can they go? True, they can each go to any of the four spaces, but if, say, they both go to the first space, in which order do they go?
Taking the two possible orders into account, we see that you are missing $$4times 3!times 4times 3=288$$ cases. Adding them back gives you the desired result.
Phrased differently: once you have placed $D_3$ there are now five available spaces for $D_4$ (since $D_4$ might go either to the left or to the right of $D_3$). thus you should have had $$3!times 4times 5times 4times 3=1440$$
answered yesterday
lulululu
43.3k25080
$endgroup$
add a comment |
$begingroup$
The second computation is missing a symmetry. Say your initial pattern is $$underline quadC_1underline quadC_2underline quadC_3underline quad$$
You then populate the spaces immediately to the right of $C_1$, and $C_2$. As:
$$underline quadC_1D_1underline quadC_2D_2underline quadC_3underline quad$$
So far so good. You still have $D_3,D_4$ to place. Where can they go? True, they can each go to any of the four spaces, but if, say, they both go to the first space, in which order do they go?
Taking the two possible orders into account, we see that you are missing $$4times 3!times 4times 3=288$$ cases. Adding them back gives you the desired result.
Phrased differently: once you have placed $D_3$ there are now five available spaces for $D_4$ (since $D_4$ might go either to the left or to the right of $D_3$). thus you should have had $$3!times 4times 5times 4times 3=1440$$
answered yesterday
lulululu
43.3k25080
$endgroup$
add a comment |
$begingroup$
The second computation is missing a symmetry. Say your initial pattern is $$underline quadC_1underline quadC_2underline quadC_3underline quad$$
You then populate the spaces immediately to the right of $C_1$, and $C_2$. As:
$$underline quadC_1D_1underline quadC_2D_2underline quadC_3underline quad$$
So far so good. You still have $D_3,D_4$ to place. Where can they go? True, they can each go to any of the four spaces, but if, say, they both go to the first space, in which order do they go?
Taking the two possible orders into account, we see that you are missing $$4times 3!times 4times 3=288$$ cases. Adding them back gives you the desired result.
Phrased differently: once you have placed $D_3$ there are now five available spaces for $D_4$ (since $D_4$ might go either to the left or to the right of $D_3$). thus you should have had $$3!times 4times 5times 4times 3=1440$$
answered yesterday
lulululu
43.3k25080
$endgroup$
The second computation is missing a symmetry. Say your initial pattern is $$underline quadC_1underline quadC_2underline quadC_3underline quad$$
You then populate the spaces immediately to the right of $C_1$, and $C_2$. As:
$$underline quadC_1D_1underline quadC_2D_2underline quadC_3underline quad$$
So far so good. You still have $D_3,D_4$ to place. Where can they go? True, they can each go to any of the four spaces, but if, say, they both go to the first space, in which order do they go?
Taking the two possible orders into account, we see that you are missing $$4times 3!times 4times 3=288$$ cases. Adding them back gives you the desired result.
Phrased differently: once you have placed $D_3$ there are now five available spaces for $D_4$ (since $D_4$ might go either to the left or to the right of $D_3$). thus you should have had $$3!times 4times 5times 4times 3=1440$$
answered yesterday
lulululu
43.3k25080
edited yesterday
answered yesterday
lulululu
43.3k25080
answered yesterday
lulululu
43.3k25080
answered yesterday
lulululu
43.3k25080
43.3k25080
add a comment |
add a comment |
$begingroup$
You can separate the two cases for the two last dogs: single dogs and double dogs.
Single dogs:
$$P(4,2)=frac4!2!=12.$$
Double dogs:
$$P(2,2)cdot C(4,1)=2cdot 4=8.$$
Hence, there are $12+8=20$ (not $4^2=16$) ways to distribute the last two dogs.
The final answer is:
$$3!cdot 20cdot 4cdot 3=1440.$$
answered yesterday
farruhotafarruhota
21.8k2842
$endgroup$
add a comment |
$begingroup$
You can separate the two cases for the two last dogs: single dogs and double dogs.
Single dogs:
$$P(4,2)=frac4!2!=12.$$
Double dogs:
$$P(2,2)cdot C(4,1)=2cdot 4=8.$$
Hence, there are $12+8=20$ (not $4^2=16$) ways to distribute the last two dogs.
The final answer is:
$$3!cdot 20cdot 4cdot 3=1440.$$
answered yesterday
farruhotafarruhota
21.8k2842
$endgroup$
add a comment |
$begingroup$
You can separate the two cases for the two last dogs: single dogs and double dogs.
Single dogs:
$$P(4,2)=frac4!2!=12.$$
Double dogs:
$$P(2,2)cdot C(4,1)=2cdot 4=8.$$
Hence, there are $12+8=20$ (not $4^2=16$) ways to distribute the last two dogs.
The final answer is:
$$3!cdot 20cdot 4cdot 3=1440.$$
answered yesterday
farruhotafarruhota
21.8k2842
$endgroup$
You can separate the two cases for the two last dogs: single dogs and double dogs.
Single dogs:
$$P(4,2)=frac4!2!=12.$$
Double dogs:
$$P(2,2)cdot C(4,1)=2cdot 4=8.$$
Hence, there are $12+8=20$ (not $4^2=16$) ways to distribute the last two dogs.
The final answer is:
$$3!cdot 20cdot 4cdot 3=1440.$$
answered yesterday
farruhotafarruhota
21.8k2842
answered yesterday
farruhotafarruhota
21.8k2842
answered yesterday
farruhotafarruhota
21.8k2842
answered yesterday
farruhotafarruhota
21.8k2842
21.8k2842
add a comment |
add a comment |
Akshat Agarwal is a new contributor. Be nice, and check out our Code of Conduct.
Akshat Agarwal is a new contributor. Be nice, and check out our Code of Conduct.
Akshat Agarwal is a new contributor. Be nice, and check out our Code of Conduct.
Akshat Agarwal is a new contributor. Be nice, and check out our Code of Conduct.
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2
$begingroup$
Is your problem arising as a consequence of a rainfall ?
$endgroup$
– Jean Marie
yesterday