Question

a salad bar offers nine choices of toppings for lettuce and how many ways can you choose five toppings

126

3024

15120

362880

Answer 1

$\displaystyle \binom{9}{5}=\dfrac{9!}{5!4!}=\dfrac{6\cdot7\cdot8\cdot9}{2\cdot3\cdot4}=126$

Answer 2

Since, a salad bar offers nine choices of toppings for lettuce.

We have to determine the number of ways in which we can choose five toppings from nine toppings.

We will use the combination formula, which states that

"
It is a formula for the number of possible combinations of r objects from a set of n objects", which is given by the formula as:

$^nC_{r}=\frac{n!}{r!(n-r)!}$

So, $^9C_{5}=\frac{9!}{5!(9-5)!}$

$^9C_{5}=\frac{9!}{5!4!}$

$^9C_{5}=\frac{9 \times 8 \times 7 \times 6 \times 5!}{5!4!}$

$^9C_{5}=\frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2}$

$= 18 \times 7$

= 126

Therefore, in 126 ways, we can choose five toppings from nine toppings for lettuce.