Math

QuestionHow many ways can you elect a president and vice-president from 8 people? Use the formula for permutations: P(n,r)=n!(nr)!P(n, r) = \frac{n!}{(n-r)!}.

Studdy Solution

STEP 1

Assumptions1. There are8 people in the club. . One person can only hold one position, either president or vice-president, not both.
3. The order of selection matters, meaning that 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

We can use the permutation formula to solve this problem. The permutation formula is(n,r)=n!(nr)!(n, r) = \frac{n!}{(n-r)!}where- nn is the total number of options, - rr is the number of options to choose, - n!n! is the factorial of nn, and- (nr)!(n-r)! is the factorial of (nr)(n-r).

STEP 3

In this problem, we have8 people and we need to choose2 (one for president and one for vice-president). So, n=8n=8 and r=2r=2.

STEP 4

Plug in the values for nn and rr into the permutation formula.
(8,2)=8!(82)!(8,2) = \frac{8!}{(8-2)!}

STEP 5

Calculate the factorial of8 and.
8!=8×7××5×4×3×2×1=40,3208! =8 \times7 \times \times5 \times4 \times3 \times2 \times1 =40,320!=×5×4×3×2×1=720! = \times5 \times4 \times3 \times2 \times1 =720

STEP 6

Substitute the calculated factorials into the permutation formula.
(8,2)=40,320720(8,2) = \frac{40,320}{720}

STEP 7

Calculate the number of different combinations.
(,2)=40,320720=56(,2) = \frac{40,320}{720} =56There are56 different combinations that can be made among eight people for the positions of president and vice-president.

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