From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?

Explanation

Total number of students = 25
No. of students to be selected for excursion party =10
`:'` There are 3 such students who decide that either all of them will join or none of then will join the excursion party.
If all three are included in the excursion party then we have to select (10-3) = 7 students out of (25-3) = 22 studetns for which no of ways of selection `=.^(22)C_(7)`
`=(lfloor22)/(lfloor7xxlfloor15) = (22 xx 21 xx 20 xx 19 xx 18 xx 17 xx 16 xx lfloor15)/(lfloor7 xx lfloor15)`
`=(22 xx 21 xx 20 xx 19 xx 18 xx 17 xx 16)/(7 xx 6 xx 5 xx 4 xx 3 xx 2 xx 1) = 170544`
If all three are not included in the excursion party then we have to select 10 students out of `(25 - 3) = 22` students for which no of ways of selection
`= .^(22)C_(10) = (lfloor22)/(lfloor10 lfloor12)`
`= (22 xx 21 xx 20 xx 19 xx 18 xx 17 xx 16 xx 15 xx 14 xx 13)/(10 xx 9 xx 8 xx 7 xx 6 xx 5 xx 4xx 3xx 2xx1)`
`= 646646`
`:.` Total no of ways of selection
`= 170544 + 646646 = 817190`
Therefore, the excursion party can be selected in 817190 ways.