Question 14 and 15 Exercise 6.2

Solutions of Question 14 and 15 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

If five distinct keys are placed on a key ring, how many different orders are possible?

The total different possible orders are: $$\dfrac{(n-1) !}{2}=\dfrac{(5-1) !}{2}=\dfrac{24}{2}=12 $$

In how many ways can $7$ people be arranged at a round table so that 2 particular persons always sit together?

Nurnber of ways in which $7$ people can be seated around a round table without any condition is $6 !$

Now, let us assume these two particular people ALWAYS sit together and let us consider them as one unit.

Number of ways in which $6$ people can be arranged around a round table is $5!$

And the two particular people can be arranged between them selves in $2 !=2$ ways.

Hence, number of ways in which $7$ people can sit around a round table

where the two people must not sit together is: $$2 \times 5 !=240$$

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