Question 4, 5 and 6, Review Exercise 6

Solutions of Question 4, 5 and 6 of Review Exercise 6 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

How mant six-digit numbers can be formed using the digits $0,2,3,4,5,7$ without repeating?

Solution.

Total possible permutations of given $6$ digits $$=6!=720$$ Permutations starting with $0$ results into $5$ digit number,
and number of such permutations is $$5!=120$$ Number of $6-$digits numbers formed $$=720-120=600$$

The numbers of ways of arranging $7$ keys in a key chain?

Solution.

Keys are to be arrange circular permutations so possible permutations are $$(7-1)!=720$$ Keys in key chain are flipable so we have to divide by $2$.
Hence possible ways of arranging keys in key chain are $$\dfrac{720}{2}=360$$

Tweleve persons are seated at a round table. Find the number of ways of their arrangement if two particular persons don't want to sit togather?

Solution. Tweleve persons can sit in a round table in $11!$ ways.
If two particular person sits togather.
We treat them as single element.
Then possible arrangements are $10!$
Arrangements when two persons are not togther $$=11!-10!=36288000$$

< Question 2 & 3