Question

From a group of 10 women and 15 men, a researcher wants to randomly select

women and men for a study in how many ways can the study group be selected?
O A 17,876
78,016,400
OG 105, 102,625
OD 00,000,000
WO

Answer 1

The total number of ways the researcher can select 5 women and 5 men for a study is 7,56,756.

Explanation:

The complete question is:

From a group of 10 women and 15 men, a researcher wants to randomly select 5 women and 5 men for a study in how many ways can the study group be selected?

Solution:

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)!)`

The number of women in the group:
`n_(w)=10`.

The number of women the researcher selects for the study,
`k_(w)=5`

Compute the total number of ways to select 5 women from 10 as follows:

`{n_(w)\choose k_(w)}=(n_(w)!)/(k_(w)!\cdot (n_(w)-k_(w))!)=(10!)/(5!\cdot (10-5)!)=(10!)/(5!* 5!)=252`

The number of men in the group:
`n_(m)=15`.

The number of men the researcher selects for the study,
`k_(m)=5`

Compute the total number of ways to select 5 men from 15 as follows:

`{n_(m)\choose k_(m)}=(n_(m)!)/(k_(m)!\cdot (n_(m)-k_(m))!)=(15!)/(5!\cdot (15-5)!)=(15!)/(5!* 10!)=3003`

Compute the total number of ways the researcher can select 5 women and 5 men for a study as follows:

`{n_(w)\choose k_(w)}* {n_(m)\choose k_(m)}=252* 3003=756756`

Thus, the total number of ways the researcher can select 5 women and 5 men for a study is 7,56,756.