Question

# Eight chairs are numbered from 1 to 8. Two women and three men wish to occupy one chair each. First women choose the chairs from amongst the chairs marked 1 to 4; and then the men select the chairs from the remaining. The number of possible arrangements is-

^{6}C_{3} × ^{4}C_{2}
^{4}P_{3} × ^{4}P_{3}
^{4}C_{2} × ^{4}P_{3}
- None of these

^{6}C_{3}×^{4}C_{2}^{4}P_{3}×^{4}P_{3}^{4}C_{2}×^{4}P_{3}Hint:

### We will first start by using the method of selecting r objects out of n objects that is for finding the ways in which we can select two chairs for women and three for men. Then we will permute the men and women among themselves.

## The correct answer is: None of these

### DETAILED SOLUTION

Now, we have been given 8 chairs which are numbered from 1 to 8. Also, it has been given that women choose the chairs from amongst the chairs marked 1 to 4, and then men select from remaining chairs.

In total there are 2 women and three men who wish to occupy one chair each.

Now, we know the number of ways of selecting r objects among n is . So, we have the ways in which we can choose two chairs among four numbered 1 to 4 is and we can arrange the women then in 2! ways. Also, we have the ways of selecting 3 chairs among the rest 6 chairs is and in them we can permute the men in 3! ways.

So, in total we have number of possible arrangements as,

2! 3!

Now, we know that =

Therefore, we have,

Total ways =

It is important to note that we have used a fact that = . This can be understood as we know that = and = . So, substituting this we have = .

### Related Questions to study

### A tea party is arranged of 16 persons along two sides of a long table with 8 chairs on each side. 4 men wish to sit on one particular side and 2 on the other side. In how many ways can they be seated ?

Whenever we face such types of problems the key point is to make special arrangements for the people who are in need of it, then arrange the remaining. Now combination comes with permutation as there are possibilities of these 8 people sitting on one side to rearrange. Thus this concept into consideration, to get through the answer.

### A tea party is arranged of 16 persons along two sides of a long table with 8 chairs on each side. 4 men wish to sit on one particular side and 2 on the other side. In how many ways can they be seated ?

Whenever we face such types of problems the key point is to make special arrangements for the people who are in need of it, then arrange the remaining. Now combination comes with permutation as there are possibilities of these 8 people sitting on one side to rearrange. Thus this concept into consideration, to get through the answer.

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