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Combinations in Committees and Groups

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5. In how many ways can a man divide 7 gifts among his 3 children if the eldest is to receive 3 gifts and the others 2 each?

6. From a group of n people, suppose that we want to choose a committee of k, k <= n, one of whom is to be designated as a chairperson.

(a) By focusing first on the choice of the committee and then on the choice of the chair, argue that there are (n choose k)?k possible choices.

(b) By focusing first on the choice of the nonchair committee members and then on the choice of the chair, argue that there are (n choose k-1)?(n - k + 1) possible choices.

(c) By focusing first on the choice of the chair and then on the choice of the other committee members, argue that there are n?(n-1 choose k-1) possible choices.

(d) Conclude from parts (a), (b), and (c) that

k?(n choose k) = (n - k + 1)?(n choose k-1) = n?(n-1 choose k-1)

(e) Use the factorial definition of (m choose r) to verify the identity in part(d).

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Solution Summary

The combinations and committees and groups are determined. The non chair committee members are analyzed.

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5. In how many ways can a man divide 7 gifts among his 3 children if the eldest is to receive 3 gifts and the others 2 each?

Total no of gifts= 7
No of ways of choosing 3 gifts for the eldest = (7 choose 3) or
= 7C 3

=(7 x 6 x 5 ) / ( 1 x 2 x 3) = 35

No of gifts remaining = 7-3 = 4

No of ways of choosing 2 gifts for the second child from the remaining 4 = (4 choose 2) or
= 4C 2

(4x3 ) / ( 1 x 2) = 6

No of gifts remaining = 4-2 = 2

No of ways of choosing 2 gifts for the third child from the remaining 2 = (2 choose 2) or
= 2C 2

(2x1 ) / ( 1 x 2) = 1

Each of the ways of choosing 3 gifts for the eldest child can be associated with each of the ways of choosing 2 gifts for the second child and with each of the ways of choosing 2 gifts for the third child.

Therefore total no of ways = 35 x6 x 1 = 210

Answer total no of ways in which a man divide 7 gifts among his 3 children if the eldest is to receive 3 gifts and the others 2 each = 210

6. From a group of n people, suppose that we want to choose a committee of k, k <= n, one of whom is to be designated as a chairperson.
Committee = k-1 non chair members + 1 chairperson

(a) By focusing first on the choice of the committee and then on the choice of the chair, ...

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