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Math  /  Discrete

QuestionIn how many ways can eight speakers be lined up at an event from a group of 17 volunteers assuming that one volunteer has already been promised the first spot? □\square Submit Question

Studdy Solution

STEP 1

1. There are 17 volunteers in total.
2. One volunteer is already assigned to the first position.
3. We need to arrange the remaining 7 positions.

STEP 2

1. Determine the number of choices for the first position.
2. Calculate the number of ways to arrange the remaining speakers.

STEP 3

Since one volunteer is already promised the first spot, there is only 1 way to fill the first position.

STEP 4

After assigning the first position, we have 16 volunteers left to fill the remaining 7 positions. We need to calculate the number of permutations of 7 volunteers from these 16.

STEP 5

The number of permutations of 7 volunteers from a group of 16 is given by the formula for permutations:
P(n,r)=n!(n−r)! P(n, r) = \frac{n!}{(n-r)!}
where n=16 n = 16 and r=7 r = 7 .

STEP 6

Substitute the values into the formula:
P(16,7)=16!(16−7)!=16!9! P(16, 7) = \frac{16!}{(16-7)!} = \frac{16!}{9!}

STEP 7

Calculate 16!9! \frac{16!}{9!} :
16×15×14×13×12×11×10 16 \times 15 \times 14 \times 13 \times 12 \times 11 \times 10

STEP 8

Perform the multiplication:
16×15=240 16 \times 15 = 240 240×14=3360 240 \times 14 = 3360 3360×13=43680 3360 \times 13 = 43680 43680×12=524160 43680 \times 12 = 524160 524160×11=5765760 524160 \times 11 = 5765760 5765760×10=57657600 5765760 \times 10 = 57657600
The number of ways to line up the eight speakers is:
57657600 \boxed{57657600}

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