Question

a state legislature consists of 21 legislators total, of which there 7 democrats, 6 republicans, and 8 independents. how many ways are there to choose:
a. a speaker, a vice - speaker, and a sergeant at arms?
b. a committee of 4 legislators?
c. a committee of 4 legislators, with no independents?
d. a committee of 6 legislators where 2 are republicans, and 2 are democrats, and 2 are independents?

Explanation:

Step1: Calculate permutations for 3 roles

Order matters: $P(21,3) = 21×20×19$
$21×20=420$, $420×19=7980$

Step2: Calculate combinations for 4 legislators

Order doesn’t matter: $C(21,4) = \frac{21!}{4!(21-4)!} = \frac{21×20×19×18}{4×3×2×1}$
$\frac{21×20×19×18}{24} = 5985$

Step3: Calculate non-Independent committee

Democrats+Republicans=13: $C(13,4) = \frac{13×12×11×10}{4×3×2×1}$
$\frac{17160}{24}=715$

Step4: Calculate mixed committee

Multiply group combinations: $C(6,2)×C(7,2)×C(8,2)$
$C(6,2)=15$, $C(7,2)=21$, $C(8,2)=28$
$15×21=315$, $315×28=8820$

Answer:

a. 7980
b. 5985
c. 715
d. 8820