JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Permutations and Combinations

The students

S_{1}, S_{2}, \dots\dots . S_{10}

are to be divided into 3 groups

A, B

and C such that each group has at least one student and the group C has at most 3 students. Then the total number of possibilities of forming such groups is ____

Answer: 31650

Solution

The groups are labelled, so a grouping is fixed by choosing the members of

C

and then assigning the rest to

A

B

. If

C

gets

r

students, the remaining

10 - r

can go to

A

B

2^{10 - r}

ways, from which the two assignments that leave

A

B

empty are removed. Summing over

r = 1, 2, 3

(1 10) (2^{9} - 2) + (2 10) (2^{8} - 2) + (3 10) (2^{7} - 2) = 10 (510) + 45 (254) + 120 (126),

which is

5100 + 11430 + 15120

, so the number of such groupings is

31650

JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Integer answer

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