Question

Ten high school students, 6 girls and 4 boys, have applied for 5 available scholarships. In how many ways can these scholarships be awarded among the applicants if a) No preference is given to any one student? b) One particular student must be awarded an assistantship? c) The group of applicants includes 6 girls and 4 boys and it is stipulated that a majority of girls must be awarded scholarships? d) At least 2 boys are awarded?e) At most 2 boys are awarded? f) What is the probability that majority of girls are awarded scholarships? d) What is the probability that at least 2 boys are awarded scholarships? e) What is the probability that at most 2 boys are awarded scholarships?

Answer 1

a) No preference is given to any one student?

total number of students = 6+4 = 10

number of scholarship = 5

hence we have to select 5 student out of 10

which can be done in 10C5 ways = 252

b) One particular student must be awarded an assistant-ship?

if one particular student is selected then we have to select 4 out of remaining 9

in 9C4 ways = 126

c) The group of applicants includes 6 girls and 4 boys and it is stipulated that a majority of girls must be awarded scholarships?

majority for girls

Y = number of girls

Y>= 3

n(Y = k) = 6Ck * 4C(5-k)

n(Y =3) + n (Y = 4) + n(Y = 5)

= 6C3 * 4C2 + 6C4 * 4C1 + 6C5

= 20 * 6 + 15 *4 + 6

=186

d) At least 2 boys are awarded?

X = number of boys who get scholarship

X >= 2

X= 2

n(X = k) = 4Ck * 6C(5-k)

4C2 * 6C3 = 6 * 20 = 120

X = 3

4C3 * 6C2 = 4*15=60

X =4

4C4 * 6C1 = 6

total = 120 + 60 + 6 = 186 ways