Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 2.47

Example 77: There are seven candidates for a post. In how many ways can a selection of four be made
amongst them, so that :
(i) 2 persons whose qualifications are below par are excluded?
(ii) 2 persons with good qualifications are included?
Solution :
(i) Excluding 2 persons, we are to select 4 out of 5 ( = 7 – 2) candidates.
Number of possible selections =^5 C 4 = 5.
(ii) In this case, 2 persons are fixed, and we are to select only 2 persons out of (7–2), i.e. 5 candidates.
Hence the required number of selection =^5 C 2 = 10.
Committee from more than one group :
Example 78: In how many ways can a committee of 3 ladies and 4 gentlemen be appointed from a
meeting consisting of 8 ladies and 7 gentlemen? What will be the number of ways if Mrs. X refuses to serve
in a committee having Mr. Y as a member?
Solution :

1st part. 3 ladies can be selected from 8 ladies in^83

C^8! 56

=3! 5!= ways and

4 gentlemen can be selected from 7 gentlemen in^74

C^7! 35

=4! 3!= ways

Now, each way of selecting ladies can be associated with each way of selecting gentlemen.
Hence, the required no. of ways = 56 × 35 = 1960.
2nd part : If both Mrs. X and Mr. Y are members of the committee then we are to select 2 ladies and 3
gentlemen from 7 ladies and 6 gentlemen respectively. Now 2 ladies can be selected out of 7 ladies in^7 C 2
ways, and 3 gentlemen can be selected out of 6 gentlemen in^6 C 3 ways.
Since each way of selecting gentlemen can be associated with each way of selecting ladies.

Hence, No. of ways^7263

C C 7! 6! 420

= × =2! 5! 3! 3!× =

Hence, the required no. of different committees, not including Mrs. X and Mr. Y
= 1960 – 420 = 1540.
Example 89 : From 7 gentlemen and 4 ladies a committee of 5 is to be formed. In how many ways can this
be done to include at least one lady? [C.U. 1984]
Possible ways of formation of a committee are :–
(i) 1 lady and 4 gentlemen (ii) 2 ladies and 3 gentlemen
(iii) 3 ladies and 2 gentlemen (iv) 4 ladies and 1 gentleman
For (i), 1 lady can be selected out of 4 ladies in^4 C 1 ways and 4 gentlemen can be selected from 7 gentlemen
in^7 C 4 ways. Now each way of selecting lady can be associated with each way of selecting gentlemen. So
1 lady and 4 gentlemen can be selected in^4 C 1 × 7 C 4 ways.
Similarly,
Case (ii) can be selected in^4 C 2 × 7 C 3 ways
Case (iii) can be selected in^4 C 3 × 7 C 2 ways