Question

A delegation of four students is to be selected from a total of 12 students. In how many ways can the delegation be selected,

(a) if all the students are equally willing?

(b) if two particular students have to be included in the delegation?

(c) if two particular students do not wish to be together in the delegation?

(d) if two particular students wish to be included together only? 

(e) if two particular students refuse to be together and two other students wish to be together only in the delegation?

Answer 1

(a) Formation of delegation means selection of 4 out of 12. Hence, the number of ways = ¹²C₄  ^{= 495.}

(b) Two particular students are already selected. Here we need to select only 2 out of the remaining 10. Hence, the number of ways = ¹⁰C₂ = 45.

(c) The number of ways in which both are selected = 45. Hence, the number of ways in which the two are not included together

= 495 – 45 = 450.

(d) There are two possible cases:

(i)  Either both are selected. In this case, the number of ways in which the selection can be made = 45.

(ii) Or both are not selected. In this case, all the four students are selected from the remaining 10 students.

This can be done in  ¹⁰C₄ = 210 ways.

Hence, the total number of ways of selection is

45 + 210 = 255

(e) We assume that students A and B wish to be selected together and students C and D do not wish to be together. 

Now there are following 6 cases: 

(i) (A, B, C) selected, (D) not selected 

(ii) (A, B, D) selected, (C) not selected 

(iii) (A, B) selected, (C, D) not selected 

(iv) (C) selected, (A, B, D) not selected 

(v) (D) selected, (A, B, C) not selected 

(vi) A, B, C, D not selected

For (i), the number of ways of selection = ⁸C₁ = 8

For (ii), the number of ways of selection = ⁸C₁ = 8

For (iii), the number of ways of selection = ⁸C₂ = 28

For (iv), the number of ways of selection = ⁸C₃ = 56

For (v), the number of ways of selection = ⁸C₃ = 56

For (vi), the number of ways of selection = ⁸C₄ = 70

Hence, total number of ways = 8 + 8 + 28 + 56 + 56 + 70 = 226