Correct Answer - Option 3 : 246
Concept:
The notation C(n, r) is the number of combinations/groups of n different things taking r at a time and is given by: \(C\left( {n,\;r} \right) = \frac{{{\rm{n}}!}}{{{\rm{r}}!\left( {{\rm{n\;}} - {\rm{\;r}}} \right)!}}\)
Calculation:
Given: 6 programmers and 4 typists, an office wants to recruit 5 people.
Here, n = 10 and r = 5.
Case 1: Group of 5 with 1 typist
1 typist can be chosen from 4 in C(4, 1) = 4 ways
4 programmers can be chosen from 6 in C(6, 4) = 15 ways
⇒ No. of ways to form a group of 5 with 1 typist = 4 × 15 = 60.
Case 2: Group of 5 with 2 typists
2 typist can be chosen from 4 in C(4, 2) = 6 ways
3 programmers can be chosen from 6 in C(6, 3) = 20 ways
⇒ No. of ways to form a group of 5 with 2 typist = 6 × 20 = 120.
Case 3: Group of 5 with 3 typists
3 typist can be chosen from 4 in C(4, 3) = 4 ways
2 programmers can be chosen from 6 in C(6, 2) = 15 ways
⇒ No. of ways to form a group of 5 with 2 typist = 4 × 15 = 60.
Case 4: Group of 5 with 4 typists
4 typist can be chosen from 4 in C(4, 4) = 1 way
1 programmers can be chosen from 6 in C(6, 1) = 6 ways
⇒ No. of ways to form a group of 5 with 2 typist = 1 × 6 = 6.
Total number of ways in forming a group of 5 with at least one typist is = 60 + 120 + 60 + 6 = 246.
