Question

10. There are fourteen juniors and twenty-three seniors in the Service Club. The club is to send four representatives to the state conference.
a. How many different ways are there to select a group of four students to attend the conference from the 37 Service Club members?
b. How many ways are there to select exactly two juniors?
c. How many ways are there to select exactly two seniors?
d. If the members of the club decide to send two juniors and two seniors, how many different groupings are possible?
e. What is the probability that two juniors and two seniors are selected to attend the conference?

Answer 1

Answer:

a) 66045

b) 91

c) 253

d) 23023

e) 0.349

Step-by-step explanation:

Number of juniors = 14

Number of seniors = 23

n = number of available choices

r = numbers of persons to be selected

a) n = 37

r = 4

number of ways  to select a group of four students = ³⁷C₄

= $\frac{37!}{4!(37-4)!}$

= $\frac{37\times36\times35\times34\times33!}{4\times3\times2\times1(33!)}$

= 66045

b) n = 14

r = 2

number of ways to select exactly two juniors = ¹⁴C₂

= $\frac{14!}{2!(14-2)!}$

= $\frac{14\times13\times12!}{2\times1(12!)}$

= 91

c) n = 23

r = 2

number of ways to select exactly two seniors = ²³C₂

= $\frac{23!}{2!(23-2)!}$

= $\frac{23\times22\times21!}{2\times1(21!)}$

= 253

d) number of ways to send two juniors and two seniors

= number of ways to select exactly two juniors × number of ways to select exactly two seniors

= ¹⁴C₂ × ²³C₂

= 91 × 253

= 23023

e) P(two juniors and two seniors are selected to attend the conference )

= number of ways to send two juniors and two seniors ÷ [number of ways  to select a group of four students]

= 23023 ÷ 66045

= 0.3486 ≈ 0.349