Page 186 - DMTH202_BASIC_MATHEMATICS_II

Unit 13: Combinations

 Number of ways of selecting at least one fruit = (4x5x6) –1 = 119 Notes
Note: There was no fruit of a diverse type, thus here n=o
n
 2 = 2 =1
0
Theorem:
Number of ways of choosing ‘r’ things from ‘n’ identical things is ‘1’.

Example: In how many ways 5 balls can be chosen from ‘12’ identical red balls?

Solution: The balls are identical, total number of manners of choosing 5 balls = 1.

Example: How many numbers of four digits can be formed with digits 1, 2, 3, 4 and 5?

Solution:
Here n = 5 [Number of digits]
And r = 4 [Number of places to be filled-up]

5
Required number is P = 5!/1! = 5 x 4 x 3 x 2 x 1
4
Self Assessment

Fill in the blanks:
6. Number of combinations of ‘n’ dissimilar things taken ‘r’ at a time, when ‘p’ particular
things are forever included = ..............................

7. Number of combination of ‘n’ dissimilar things, taken ‘r’ at a time, when ‘p’ particular
things are forever to be excluded = ............................. .

13.3 Binomial Coefficients

You must be familiar with expressions like a + b, p + q, x + y, all consisting of two terms. This is
why they are called binomials. You also know that a binomial expansion refers to the expansion
of a positive integral power of such a binomial.

4
5
3 2
5
2 3
4
5
For instance, (a + b) = a + 5a b + 10a b + 10a b + 5ab + b is a binomial expansion. Consider
coefficients 1, 5, 10, 10, 5, 1 of this expansion. In particular, let us consider the coefficient 10, of
3 2
a b in this expansion. We can get this term by selecting a from 3 of the binomials and b from the
remaining 2 binomials in the product (a + b) (a + b) (a + b) (a + b) (a + b). Now, a can be chosen
in C(5, 3) ways, i.e., 10 ways. This is the way each coefficient arises in the expansion.
r
n
The same argument can be extended to get the coefficients of a b n – r in the expansion of (a + b) .
n
From the n factors in (a + b) , we have to select r for a and the remaining (n – r) for b. This can be
n
r
done in C(n, r) ways. Thus, the coefficient of a b n “ r in the expansion of (a + b) is C(n, r).
!
b will
r n – r
Caution In view of the fact that C(n, r) = C(n, n – r), the coefficients of a b and a n – r r
be the same.
r can only take the values 0, 1, 2, …, n. We also see that C(n, 0) = C(n, n) = 1 are the coefficients of
a and b . Hence we have established the binomial expansion.
n
n
b + … + C(n, r) a
n
n
b + … + b .
(a + b) = a + C(n, 1) a n – 1 b + C(n, 2)a n – 2 2 n – r r n
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