Question

from a group of 18 smokers and 22 nonsmokers, a researcher wants to randomly select 7 smokers and 7 nonsmokers for a study. in how many ways can the study group be selected?
how many ways can the group be selected?
a. 27,354,056,970,240
b. 5,427,392,256
c. 347,582,329,473,600
d. 137,864,447,130,010,000

Explanation:

Step1: Calculate smoker combinations

We use the combination formula $C(n,k)=\frac{n!}{k!(n-k)!}$ for 18 smokers choose 7:
$C(18,7)=\frac{18!}{7!(18-7)!}=\frac{18!}{7!11!}=31824$

Step2: Calculate nonsmoker combinations

Use the combination formula for 22 nonsmokers choose 7:
$C(22,7)=\frac{22!}{7!(22-7)!}=\frac{22!}{7!15!}=170544$

Step3: Multiply the two results

Total ways = $C(18,7) \times C(22,7)$
$31824 \times 170544 = 5,427,392,256$

Answer:

B. 5,427,392,256