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Find the number of selection of r things out of n different things such that none of particular things are selected at a time. (Topic:Permutation And Combination)

Find the number of selection of r things out of n different things such that none of  particular things are selected at a time.
(Topic:Permutation And Combination)
 

Grade:11

2 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
9 years ago
Hello student,
Please find the answer to your question below
The number of permutations of n dissimilar things taken r at a time when k(< r) particular things always occur is [(n-k)P(r-k)] .[ rPk]
The easy answer is that there are(n−kPr-k)timesr!such permutations. Once you have chosen ther−kobjects from then−k, to join thekthat must be selected, the resulting collection can be lined up inr!different orders.
Explanation:
Say thekspecial objects we must take are red, and numbered1tok. Suppose the rest of the objects are blue.
Imagine also that ourrobjects will be placed intorconsecutive slots. We grab and line upr−k blue objects from then−kblue objects available. This can be done in P(n-k, r-k) ways.
Then we make an ordered selection (permutation) ofkslots from theravailable. Thekred objects will be placed in the chosen slots, with red1going into the first slot in the permutation, red2in the second slot in the permutation, and so on. The permutation ofkslots chosen fromrcan be done inP(r,k)ways. Then the empty slots are filled, from left to right, with our permutation ofr−k objects taken fromn−k.
.
SHAIK AASIF AHAMED
askIITians Faculty 74 Points
9 years ago
Hello student,
Please find the answer to your question below
The number of permutations of n dissimilar things taken r at a time when k(< r) particular things always occur is [(n-k)P(r-k)] .[ rPk]
The easy answer is that there are(n−kPr-k)timesr!such permutations. Once you have chosen ther−kobjects from then−k, to join thekthat must be selected, the resulting collection can be lined up inr!different orders.
Explanation:
Say thekspecial objects we must take are red, and numbered1tok. Suppose the rest of the objects are blue.
Imagine also that ourrobjects will be placed intorconsecutive slots. We grab and line upr−k blue objects from then−kblue objects available. This can be done in P(n-k, r-k) ways.
Then we make an ordered selection (permutation) ofkslots from theravailable. Thekred objects will be placed in the chosen slots, with red1going into the first slot in the permutation, red2in the second slot in the permutation, and so on. The permutation ofkslots chosen fromrcan be done inP(r,k)ways. Then the empty slots are filled, from left to right, with our permutation ofr−k objects taken fromn−k.
.

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