Sol. Formation of delegation means selection of 4 out of 12. Hence the number of ways = 12 C 4 = 495

If two particular students are already selected. Here we need to select only 2 out of the remaining 10. Hence the number of ways = 10 C 2 = 45

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

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 ten students.

This can be done in 10 C 4 = 210 ways.

Hence the total number of ways of selection = 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, not selected.

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

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

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

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

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

For (i) the number of ways of selection = 8 C 1 = 8

For (ii) the number of ways of selection = 8 C 1 = 8

For (iii) the number of ways of selection = 8 C 2 = 28

For (iv) the number of ways of selection = 8 C 3 = 56

For (v) the number of ways of selection = 8 C 3 = 56

For (vi) the number of ways of selection = 8 C 4 = 70

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