Question

The student council at a college is made up of four ​freshmen, five ​sophomores, six ​juniors, and seven seniors. A yearbook photographer would like to line up two council members from each class for a picture. How many different pictures are possible if each group of classmates stands​ together?

Answer 1

Answer:

7257600

Step-by-step explanation:

Number of freshmen in the student council= 4

Number of sophomores in the student council= 5

Number of juniors in the student council= 6

Number of seniors in the student council= 7

Ways of choosing council members

⁴C₂×⁵C₂×⁶C₂×⁷C₂

$^4C_2=\frac{4!}{(4-2)!2!}\\=\frac{24}{4}=6\\\\^5C_2=\frac{5!}{(5-2)!2!}\\=\frac{120}{12}=10\\\\^6C_2=\frac{6!}{(6-2)!2!}\\=\frac{720}{48}=15\\\\^7C_2=\frac{7!}{(7-2)!2!}\\=\frac{5040}{240}=21$

⁴C₂×⁵C₂×⁶C₂×⁷C₂=6×10×15×21=18900

Ways of lining up the four classes=4!=1×2×3×4=24

Ways of lining up members of each class=2⁴=2×2×2×2=16

Pictures are possible if each group of classmates stands​ together

⁴C₂×⁵C₂×⁶C₂×⁷C₂×4!×2⁴

=18900×24×16

=7257600