Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 2.49

Now,^7 C 5 = 7.6 = 42 ways.
(ii) Here we have the following possible cases :
(a) 4 questions from first 5 questions (say, group A), then remaining 4 questions from the balance of 5
questions (say, group B).
(b) Again 5 questions from group A, and 3 questions from group B.
For (a), number of choice is^5 C 4 × 5 C 4 = 5 × 5 = 25
For (b) number of ways is^5 C 5 × 5 C 3 = 1 × 10 = 10.
Hence, Required no. of ways = 25 + 10 = 35.
Example 84 : Given n points in space, no three of which are collinear and no four coplanar, for what value
of n will the number of straight lines be equal to the number of planes obtained by connecting these
points?
Solution :
Since no three points, are collinear, the number of lines = number of ways in which 2 points can be selected
out of n points

n ( )

2

C n n 1

2

= = − lines

Again since three non-collinear points define a space and no four of the points are coplaner ; the number
of planes = number of ways in which 3 points can be selected out of n points.

n ( )( )

3

= C =n n-1 n- 2

6

Now, we have ( ) ( ( ))

n n 1 (^1) n n 1 n 2 ;
2 6

= − = − −

or, 6 = 2 (n – 2) Hence, n = 5

SELF EXAMINATION QUESTIONS

1. In an examination paper, 10 questions are set. In how many different ways can you choose 6 questions
   to answer. If however no. 1 is made compulsory in how many ways can you select to answer 6 questions
   in all? [Ans. 210, 126]


2. Out of 16 men, in how many ways a group of 7 men may be selected so that :
   (i) particular 4 men will not come,
   (ii) particular 4 men will always come? [Ans. 792 ; 220]


3. Out of 9 Swarjists and 6 Ministerialists, how many different committees can be formed, each consisting
   of 6 Swarajists and Ministerialists? [Ans. 1680]


4. A person has got 15 acquaintances of whom 10 are relatives. In how many ways may be invite 9
   guests so that 7 of them would be relatives? [Ans. 1200]


5. A question paper is divided in three groups A, B and C each of which contains 3 questions, each of 25
   marks. One examinee is required to answer 4 questions taking at least one from each group. In how
   many ways he can choose the questions to answer 100 marks [ [Ans. 81]

[hints : (^) (^3 C C C 1 ×^31 ×^32 )+(^3 C C C 1 ×^32 ×^31 )+(^3 C C C 2 ×^31 ×^31 ) etc.]