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.
Question 14
If five distinct keys are placed on a key ring, how many different orders are possible?
Solution
The total different possible orders are: $$\dfrac{(n-1) !}{2}=\dfrac{(5-1) !}{2}=\dfrac{24}{2}=12 $$
Question 15
In how many ways can $7$ people be arranged at a round table so that 2 particular persons always sit together?
Solution
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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