A delegation of four students is to be selected from all 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 in the delegation?

Explanation

(i) Formation of delegation means selection of 4 out of 12.
Hence, the number of ways is `.^(12)C_(4)=495`.
(ii) If two particular students are already selected, we need to select only 2 out of the remaining 10. Hence, the number of ways is `.^(10)C_(2)=45`.
(iii) The number of ways in which both are selected is 45. Hence, the number of ways in which the two are not included together is 495-45=450.
(iv) There are two possible cases :
(a) Both are selected. In this case, the number of ways in which the selection can be made is 45.
(b) Both are not selected. In this case, all the 4 students are selected from the remaining 10 students. This can be done in `.^(10)C_(4)=210` ways.
Hence, total number of ways of selection is 45+210=255.
(v) We asssume that students A and B wish to be selected together and students C and D do not wish to be together.

The total number of ways is
`s_(1)-s_(3)+s_(2)-s_(4)=45-1+210-28`
=216