# There are 11 students who are willing to help with a charity event, but only 5 students are needed. Nobody has been chosen yet. How many different groups could be selected if Kate is willing to do...

There are 11 students who are willing to help with a charity event, but only 5 students are needed. Nobody has been chosen yet. How many different groups could be selected if Kate is willing to do it only if Sandra also does it?

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Total number of students are 11. In the selection of 5 students there is restriction. Following cases are possible

(i) In group there will Kate and Sandra.

(ii) In group there will neither Kate nor Sandra.

(iii) In group there will Sandra but no Kate.

Thus

Number of possible selection in (i) are =`^9C_3=(9!)/(3!xx6!)`

`=84`

Number of possible selection in (ii) are =`^9C_5=(9!)/(5!xx4!)`

`=126`

Number of possible selection in (iii) are =`^10C_5=(10!)/(5!xx5!)`

`=252`

Thus total possible selection of 5 students from group of 11 students are =84+126+252 = 462.

There are total 11 students and required to form groups of 5 students.

There are two case

(i) There are are Kate and Sandra

(ii) There will no Kate and Sandra.

(i) In this case Kate and Sandra there , we need only three students

from total 9 students. Thus possible selections are

`^9C_3=(9!)/(3!xx(9-3)!)`

`=84`

(ii) In this case there will no Kate and Sandra , so we need to select

5 students from total 9 students.

Thus possible selections are

`^9C_5=(9!)/(5!(9-4)!)`

`=126`

Thus total possible selection of 5 students from group of 11 atudents are=126+84

=210