Find the number of permutations of n dissimilar things taken r at a time.
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The easy answer is that there are (n−kr−k)(n−kr−k) times r!r! such permulations. Once you have chosen the r−kr−k objects from the n−kn−k, to join the kk that must be selected, the resulting collection can be lined up in r!r! different orders.
Remark (in response to a question in a comment): In a long list of "types" of problems, the web site mentioned in the comment offers P(n−k,r−k)P(r,k)P(n−k,r−k)P(r,k) as the answer to the question.
First we express our answer above in terms of factorials. It is
(n−k)!(r−k)!(n−r)!r!.(1)(1)(n−k)!(r−k)!(n−r)!r!.
The answer offered in the link, in terms of factorials,
(n−k)!(n−r)!⋅r!(r−k)!.(2)(2)(n−k)!(n−r)!⋅r!(r−k)!.
Clearly the two answers are the same.
We describe the reasoning behind Formula (2)(2). Say the kk special objects we must take are red, and numbered 11 to kk. Suppose the rest of the objects are blue.
Imagine also that our rr objects will be placed into rr consecutive slots. We grab and line up r−kr−k blue objects from the n−kn−k blue objects available. This can be done in P(n-k, r-k)$ ways.
Then we make an ordered selection (permutation) of kk slots from the rravailable. The kk red objects will be placed in the chosen slots, with red 11 going into the first slot in the permutation, red 22 in the second slot in the permutation, and so on. The permutation of kk slots chosen from rr can be done in P(r,k)P(r,k) ways. Then the empty slots are filled, from left to right, with our permutation of r−kr−kobjects taken from n−kn−k. That gives the product described in the link.
A final comment: The list of "types" mentioned is quite long, with a special formula for each type. Presumably one is invited to memorize all of these. I believe that it is much better to have a limited number of basic strategies than to try to remember a large number of "canned" formulas. One develops such strategies by working through many diverse problems.
Remark (in response to a question in a comment): In a long list of "types" of problems, the web site mentioned in the comment offers P(n−k,r−k)P(r,k)P(n−k,r−k)P(r,k) as the answer to the question.
First we express our answer above in terms of factorials. It is
(n−k)!(r−k)!(n−r)!r!.(1)(1)(n−k)!(r−k)!(n−r)!r!.
The answer offered in the link, in terms of factorials,
(n−k)!(n−r)!⋅r!(r−k)!.(2)(2)(n−k)!(n−r)!⋅r!(r−k)!.
Clearly the two answers are the same.
We describe the reasoning behind Formula (2)(2). Say the kk special objects we must take are red, and numbered 11 to kk. Suppose the rest of the objects are blue.
Imagine also that our rr objects will be placed into rr consecutive slots. We grab and line up r−kr−k blue objects from the n−kn−k blue objects available. This can be done in P(n-k, r-k)$ ways.
Then we make an ordered selection (permutation) of kk slots from the rravailable. The kk red objects will be placed in the chosen slots, with red 11 going into the first slot in the permutation, red 22 in the second slot in the permutation, and so on. The permutation of kk slots chosen from rr can be done in P(r,k)P(r,k) ways. Then the empty slots are filled, from left to right, with our permutation of r−kr−kobjects taken from n−kn−k. That gives the product described in the link.
A final comment: The list of "types" mentioned is quite long, with a special formula for each type. Presumably one is invited to memorize all of these. I believe that it is much better to have a limited number of basic strategies than to try to remember a large number of "canned" formulas. One develops such strategies by working through many diverse problems.
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