Question

A group of 25 people are going to run a race. The top 8 finishers advance to the finals. Determine the number of possible outcomes.

Answer 1

Answer:

n[total]=25

n[top finishers]=8

the number of possible outcome

=$\frac{25 \times 24 \times 23 \times 22 \times 21 \times 20 \times 19 \times 18}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 1081575$outcomes

Answer 2

Answer:

$\large\boxed{ \boxed{\rm 1081575 \: outcomes}}$

Step-by-step explanation:

to determine the number of possible outcomes we can consider Combination given by

$\displaystyle \binom{n}{r} = \frac{n!}{r!(n - r)!}$

where:

• n is the total number of objects in the set
• r is the number of choosing objects from the set

so let n and r be 25 and 8 respectively thus

substitute:

$\displaystyle \binom{25}{8} = \frac{25!}{8!(25- 8)!}$

simplify parentheses:

$\displaystyle \binom{25}{8} = \frac{25!}{8! \times 17!}$

rewrite numeratorr:

$\rm \displaystyle \binom{25}{8} = \frac{25 \times 24 \times 23 \times 22 \times 21 \times 20 \times 19 \times 18 \times 17!}{8! \times 17!}$

reduce fraction:

$\rm \displaystyle \binom{25}{8} = \frac{25 \times 24 \times 23 \times 22 \times 21 \times 20 \times 19 \times 18 }{8! }$

simplify numerator and denominator:

$\rm \displaystyle \binom{25}{8} = \frac{43609104000}{40320 }$

simplify division:

$\rm \displaystyle \binom{25}{8} = 1081575$

hence,

the number of outcomes is 1081575