Answer to Question #121086 in Discrete Mathematics for Nayra

Question

Combinations:
(a) A class contains 10 students with 6 men and 4 women. Find the number of ways to:
i. Select a 4-member committee from the students
ii. Select a 4-member committee with 2 men and 2 women
iii. Elect a president, vice president, and secretary
(b) A box contains 8 blue socks and 6 red socks. Find the number of ways two socks can be drawn
from the box if:
i. They can be any color.
ii. They must be the same color.

Answer 1

a) A class contains 10 students with 6 men and 4 women.

i) The number of ways to select a 4-member committee from the students  = $C^{10}_4 = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210$ .

ii) The number of ways to select a 4-member committee with 2 men and 2 women = $C^6_2 C^4_2 = \frac{6 \times 5}{2 \times 1} \ \frac{4 \times 3}{2 \times 1} = 15 \times 6 = 90$ .

b) A box contains 8 blue socks and 6 red socks.

Total socks = 14.

i) The number of ways two socks can be drawn from the box if they can be any color = $C^{14}_2 = \frac{14 \times 13}{ 2 \times 1} = 91$ .

ii) The number of ways two socks can be drawn from the box if they must of same color = Number of ways that they must of blue color + Number of ways that they must of red color = $C^8_2 + C^6_2 = \frac{8 \times 7}{ 2 \times 1} + \frac{6 \times 5}{ 2 \times 1} = 21 + 15= 36$ .