Explore BrainMass

Explore BrainMass

    Combinations in Committees and Groups

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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).

    © BrainMass Inc. brainmass.com March 4, 2021, 6:10 pm ad1c9bdddf
    https://brainmass.com/math/combinatorics/combinations-committees-group-32735

    Attachments

    Solution Preview

    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, ...

    Solution Summary

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

    $2.49

    ADVERTISEMENT