Math

QuestionHow many ways can you choose a president and vice-president from 9 people? Use the formula for permutations: P(n,r)=n!(nr)!P(n, r) = \frac{n!}{(n-r)!}.

Studdy Solution

STEP 1

Assumptions1. There are9 people to choose from. The president and vice-president are distinct positions, so the same person cannot hold both positions3. The order of selection matters, i.e., choosing person A as president and person B as vice-president is different from choosing person B as president and person A as vice-president

STEP 2

This is a permutation problem, as the order of selection matters. The formula for permutations is(n,r)=n!(nr)!(n, r) = \frac{n!}{(n-r)!}where n is the total number of items, r is the number of items to choose, and "!" denotes factorial.

STEP 3

In this problem, we have n =9 (the total number of people) and r =2 (the number of positions to fill president and vice-president). Plug these values into the permutation formula(9,2)=9!(92)!(9,2) = \frac{9!}{(9-2)!}

STEP 4

Calculate the factorial of9 and7 (since9-2=7):
9!=9×8×7×6××4×3×2×19! =9 \times8 \times7 \times6 \times \times4 \times3 \times2 \times17!=7×6××4×3×2×17! =7 \times6 \times \times4 \times3 \times2 \times1

STEP 5

Substitute these values back into the permutation formula(9,2)=9!7!=9×8×7××5×4×3×2×17××5×4×3×2×1(9,2) = \frac{9!}{7!} = \frac{9 \times8 \times7 \times \times5 \times4 \times3 \times2 \times1}{7 \times \times5 \times4 \times3 \times2 \times1}

STEP 6

implify the fraction by cancelling out the common factors(9,2)=9×8=72(9,2) =9 \times8 =72So, there are72 different president/vice-president combinations possible.

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