In how many can a committee of 5 be formed from 4 professors and 6 students so as to include at least 2 professors ?

In how many ways 4 cards can be selected from a full pack of 52 cards , so as to include at least 2 aces ?

In how many ways a committee of 6 can be formed from a group of 10 doctors of whom 6 are ladies and 4 are gents so that one particular lady doctor must be included in the committee.

A box contains 12 lamps of which 5 are defective . In how many ways can a sample of 6 be selected at random from the box so as to include at most 3 defective lamps ?

From 8 boys and 5 girls , how many different selections of 5 members can be made so as to include at least one boy and one girl ?

In how many ways can 3 mangoes be selected from 10 mangoes so as to always include a particular mango?

In how many ways can be 9 man be selected from 15 men so as always to Include 3 particular man.

In how many ways can 6 boys form a ring ?

In how many ways can a team of 11 players be formed out of 25 players, if 6 out of them are always to be included and 5 always to be excluded a. 2020 b. 2002 c. 2008 d. 8002

Find the number of committees of 5 members that can be formed from 6 gentlemen and 4 ladies if each committee has at least one lady and two gentlemen .

A committee of 12 is to be formed from nine women and eight men. In how many ways can this be done if at least five women have to be included in a committee? In how many of these committees a. the women hold majority? b. the men hold majority?