Page 259 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Quantitative Techniques – I

Notes 3. the total number of ways of simultaneous throwing of 2 coins and a die is equal to
2 2 6 = 24.

Example:
1. In how many ways can the letters of the word EDUCATION be arranged?
2. In how many ways can the letters of the word STATISTICS be arranged?

3. In how many ways can 20 students be allotted to 4 tutorial groups of 4, 5, 5 and 6 students
respectively?
4. In how many ways 10 members of a committee can be seated at a round table if (a) they can
sit anywhere (b) president and secretary must not sit next to each other?
Solution:
1. The given word EDUCATION has 9 letters. Therefore, number of permutations of 9 letters
is 9! = 3,62,880.
2. The word STATISTICS has 10 letters in which there are 3S’s, 3T’s, 2I’s, 1A and 1C. Thus, the
required number of permutations = 50,400.
3. Required number of permutations = 9,77,72,87,522
4. (a) Number of permutations when they can sit anywhere = (10-1)!= 9! = 3,62,880.
(b) We first find the number of permutations when president and secretary must sit
together. For this we consider president and secretary as one person. Thus, the
number of permutations of 9 persons at round table = 8! = 40,320.

The number of permutations when president and secretary must not sit together = 3,62,880 –
40,320 = 3,22,560.

12.3.3 Combination

When no attention is given to the order of arrangement of the selected objects, we get a
n
combination. We know that the number of permutations of n objects taking r at a time is P .
r
Since r objects can be arranged in r! ways, therefore, there are r! permutations corresponding to
one combination. Thus, the number of combinations of n objects taking r at a time, denoted by
n P n!
n n n C r
C , can be obtained by dividing P by r!, i.e., r
r r r! r! n r !
Note:

n
n
1. Since C n C , therefore, C is also equal to the combinations of n objects taking
r n r r
(n - r) at a time.
2. The total number of combinations of n distinct objects taking 1, 2, ...... n respectively, at a
n
time is C n C ...... n C 2 n 1 .
1 2 n

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