Question

Anja is choosing her extracurricular activities for the year. She can choose one sport to play and one instrument to learn using the list below:

Sports: softball, basketball, tennis, or swimming
Instruments: guitar, piano, or clarinet​. How many combinations are possible?

Answer 1

12

Explanation:

Answer 2

The number of possible combinations of sports and instrument that Anja can select is 12.

Explanation:

In mathematics, the procedure to select k items from n distinct items, without replacement, is known as combinations.

The formula to compute the combinations of k items from n is given by the formula:

`{n\choose k}=(n!)/(k!\cdot(n-k)!)`

It is said that Anja can choose one sport to play and one instrument to learn using the list below:

Sports: softball, basketball, tennis, or swimming

Instruments: guitar, piano, or clarinet​.

There 4 options for sports and 3 for an instrument.

Compute the number of ways to select one sport to play as follows:

`n (S)={4\choose 1}=(4!)/(1!\cdot(4-1)!)=(4!)/(3!)=(4*3!)/(3!)=4`

Compute the number of ways to select one instrument to learn as follows:

`n(I)={3\choose 1}=(3!)/(1!\cdot(3-1)!)=(3!)/(2!)=(3*2!)/(2!)=3`

Compute the number of possible combinations of sports and instrument that Anja can select as follows:

Total number of possible combinations = n (S) × n (I)

`=4* 3\\=12`

Thus, the number of possible combinations of sports and instrument that Anja can select is 12.