FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 2.45- Show that the number of ways in which 16 different books can be arranged on a shelf so that two
particular books shall not be together is 14 (15)! - In how many ways can the letters of the word MONDAY be arranged? How many of them begin with
M? How many of them do not begin with M but end with Y? [Ans. 720, 120, 96] - In how many ways can 5 boys form a ring? [Ans. 24]
- In how many ways 5 different beads be strung on a necklace? [Ans. 12]
2.6.2. COMBINATION
Definition :
The different groups or collection or selections that can be made of a given set of things by taking some or
all of them at a time, without any regard to the order of their arrangements are called their combinations.
Thus the combinations of the letters a, b, c, taking one, two or three at a time are respectively.
a ab abc
b bc
c ca
Combinations of things all different :
To find the number of combinations of n different things taken r (r ≤ n) at a time, i.e., to find the value of nCr.
Let X denote the required number of combinations, i.e., X = nCr.
Now each combination contains r different things which can be arranged among themselves in r! ways.
So X combinations will produce X. r! which again is exactly equal to the number of permutations of n
different things taken r at a time, i.e., nPr
Hence, X × r! = nPr
( )
nPr n!
X = =r! r! n- r! Since, Pnr (n r !n!)
= −(^)
nCr n!
∴ =r!(n r)!−
Cor. nC 1 = n taking r = 1
nCn = 1, taking r = n
nC 0 = 1, taking r = 0
Restricted Combination :
To find the number of combinations of n different things taken r at a time, with the following restrictions :
(i) p particular things always occur ; and
(ii) p particular things never occur.
(i) Let us first consider that p particular things be taken always ; thus we have to select (r – p) things from
(n – p), which can be done in (n p−)C(r p−) ways.
(ii) In this case, let those p things be rejected first, then we have to select r things from the remaining (n –
p) things, which can be done in n–pCr ways.