Exercise: counting committees 0.0/2.0 puntos (calificable) we start with a pool of n people. a chaired committee consists of k≥1 members, out of whom one member is designated as the chairperson. the expression k(nk) can be interpreted as the number of possible chaired committees with k members. this is because we have (nk) choices for the k members, and once the members are chosen, there are then k choices for the chairperson. thus, c=∑k=1nk(nk) is the total number of possible chaired committees of any size. find the value of c (as a function of n ) by thinking about a different way of forming a chaired committee: first choose the chairperson, then choose the other members of the committee. the answer is of the form c=(α+nβ)2γn+δ. what are the values of α , β , γ , and δ ?

Question

Exercise: counting committees 0.0/2.0 puntos (calificable) we start with a pool of n people. a chaired committee consists of k≥1 members, out of whom one member is designated as the chairperson. the expression k(nk) can be interpreted as the number of possible chaired committees with k members. this is because we have (nk) choices for the k members, and once the members are chosen, there are then k choices for the chairperson. thus, c=∑k=1nk(nk) is the total number of possible chaired committees of any size. find the value of c (as a function of n ) by thinking about a different way of forming a chaired committee: first choose the chairperson, then choose the other members of the committee. the answer is of the form c=(α+nβ)2γn+δ. what are the values of α , β , γ , and δ ?

Answer 1

To find the values of α, β, γ, and δ, we need to express the total number of possible chaired committees, c, as a function of n by considering a different way of forming a chaired committee.

First, let's count the number of possible committees using the alternate method:
1. Choose the chairperson: We have n choices for the chairperson.
2. Choose the other members of the committee: Once the chairperson is chosen, we need to select k-1 members from the remaining n-1 people. This can be done in (n-1)C(k-1) ways.

Therefore, the total number of committees is given by:

c = Σ(k=1 to n) [n * (n-1)C(k-1)]

To simplify this expression, we can use the binomial coefficient identity: (n-1)C(k-1) = nCk.

c = Σ(k=1 to n) [n * nCk]

Now, let's expand this expression using the binomial theorem:

c = n * Σ(k=1 to n) [(nCk)]

The sum Σ(k=1 to n) [(nCk)] represents the sum of the binomial coefficients for all possible values of k from 1 to n. We know that the sum of all binomial coefficients equals 2^n.

c = n * 2^n

Now we can rewrite c in the required form c = (α + nβ)^(2γn + δ).

Comparing the two expressions, we find:

α + nβ = n (This implies α = 0 and β = 1)

2γn + δ = 1 (This implies γ = 1/2 and δ = 0)

Therefore, the values of the constants in the expression for c are:
α = 0
β = 1
γ = 1/2
δ = 0

Hence, c = (n + n)^(2(1/2)n + 0) = (2n)^(n)