Question

There are 5 professors and 16 students find the number of ways in which a committee of 5 can be formed as to include (1) exactly 3 professors (2) at least 3 professors (3) at the most 3 professors?

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Answer 1

Answer:

(1) 1200 ways

(2) 1281 ways

(3) 20,268 ways

Step-by-step explanation:

The total number of members is, 5 professors + 16 students = 21 member.

A committee of 5 member is to formed.

The number of ways to select 5 members from 21 is,

$N={21\choose 5}=\frac{21!}{5!(21-5)!} =20349$

(1)

Number of ways to select 5 members such that there are exactly 3 professors is = Number of ways to select 3 professors + Number of ways to select 2 students

= n (3 professor) + n (2 student)

$={5\choose 3}+{16\choose 2}\\=\frac{5!}{3!(5-3)!}\times\frac{16!}{2!(16-2)!} \\=10\times120\\=1200$

Thus, there are 1200 ways to select 5 members such that there are exactly 3 professors in the committee.

(2)

Number of ways to select 5 members such that there are at least 3 professors is = Total number of ways to select 5 members - Number of ways to select less than 3 professors

= N - n (less than 3 professor)

= N - n (2 professor) - n (1 professor) - n (0 professor)

$=20349-[{5\choose 2}\times{16\choose 3}]-[{5\choose 1}\times{16\choose 4}]-[{5\choose 0}\times{16\choose 5}]\\=20349-5600-9100-4368=1281$

Thus, there are 1281 ways to select 5 members such that there are at least 3 professors in the committee.

(3)

Number of ways to select 5 members such that there are at most 3 professors is = Total number of ways to select 5 members - Number of ways to select more than 3 professors

= N - n (more than 3 professor)

= N - n (4 professor) - n (5 professor)

$=20349-[{5\choose 4}\times{16\choose 1}]-[{5\choose 5}\times{16\choose 0}]\\=20349-80-1\\=20268$

Thus, there are 20,268 ways to select 5 members such that there are at most 3 professors in the committee.

Answer 2

Answer:

Step-by-step explanation:

The total number of members is, 5 professors + 16 students = 21 member.

A committee of 5 member is to formed.

The number of ways to select 5 members from 21 is,

(1)

Number of ways to select 5 members such that there are exactly 3 professors is = Number of ways to select 3 professors + Number of ways to select 2 students

= n (3 professor) + n (2 student)

Thus, there are 1200 ways to select 5 members such that there are exactly 3 professors in the committee.

(2)

Number of ways to select 5 members such that there are at least 3 professors is = Total number of ways to select 5 members - Number of ways to select less than 3 professors

= N - n (less than 3 professor)

= N - n (2 professor) - n (1 professor) - n (0 professor)

Thus, there are 1281 ways to select 5 members such that there are at least 3 professors in the committee.

(3)

Number of ways to select 5 members such that there are at most 3 professors is = Total number of ways to select 5 members - Number of ways to select more than 3 professors

= N - n (more than 3 professor)

= N - n (4 professor) - n (5 professor)

Thus, there are 20,268 ways to select 5 members such that there are at most 3 professors in the committee.