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प्रश्न
There are 5 teachers and 20 students. Out of them a committee of 2 teachers and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees a particular teacher is included?
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उत्तर
The number of teachers = 5
Number of students = 20
The number of ways of selecting 2 teachers from 5 teachers is
= 5C2 ways.
= `(5!)/(2! xx (5 - 2)!)`
= `(5!)/(2! xx 3!)`
= `(5 xx 4 xx 3!)/(2! xx 3!)`
= `(5 xx 4)/(2 xx 1)`
= 10 ways
The number of ways of selecting 3 students from 20 students is
= 20C3
= `(20!)/(3! xx (20 - 3)!)`
= `(20!)/(3! xx 17!)`
= `(20 xx 19 xx 18 xx 17!)/(3! xx 17!)`
= `(20 xx 19 xx 18)/(3 xx 2 xx 1)`
= 20 × 19 × 3
= 1140 ways
∴ The total number of selection of the committees with 2 teachers and 5 students is
= 10 × 1140
= 11400
A particular teacher is included.
Given a particular teacher is selected.
Therefore, the remaining 1 teacher is selected from the remaining 4 teachers.
Therefore, the number of ways of selecting 1 teacher from the remaining 4 teacher
= 4C1 ways
= 4 ways
The number of ways of selecting 3 students from 20 students = 20C3
= `(20!)/(3!(20 - 3)!)`
= `(20!)/(3! xx 17!)`
= `(20 xx 19 xx 18 xx 17!)/(3! xx 17!)`
= `(20 xx 19 xx 18)/(3!)`
= `(20 xx 19 xx 18)/(3 xx 2 xx 1)`
Hence the required number of committees
= 1 × 4 × 1140
= 4560
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