Question

If n is the number of necklaces which can be formed using 17 identical pearls and two identical diamonds and similarly m is the number of necklaces which can be formed using 17 identical pearls and different diamonds , then
A) n = 9
B) m=18
C) n=18
D) m = 9

Answer 1

Hint:
In the first case we are given that we have 17 identical pearls and 2 identical diamonds and it is nothing other than splitting it into two groups and in the second case one diamond is placed in one way and the other can be placed in 9 ways.

Complete step by step solution:
In the first case we are given that we have 17 identical pearls and 2 identical diamonds
This is the same as splitting 17 identical things into two groups.
$ \Rightarrow \dfrac{{17 + 1}}{2} = \dfrac{{18}}{2} = 9$
Hence the number of necklaces which can be made using 17 identical pearls and 2 identical diamonds is 9
$ \Rightarrow n = 9$
In the second case, the first diamond can be placed in one way and the second diamond can be placed in 9 different ways.
Hence the number of of necklaces which can be made using 17 identical pearls and 2 different diamonds is 9
$ \Rightarrow m = 9$
Hence both m and n are 9

Hence both options b and d are correct.

Note:
In the second case, there is no effect if the diamonds are different as the necklace can be flipped. Hence it has the same value as the number of necklaces formed by using identical diamonds.