Maths-

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Easy

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

Maths-

### 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 ?</span

DETAILED SOLUTION:

There are 16 people for the tea party.

People sit along a long table with 8 chairs on each side.

Out of 16, 4 people sit on a particular side and 2 sit on the other side.

Therefore first we will make sitting arrangements for those 6 persons who want to sit on some specific side.

Now we have remaining (16 - 6) =10 persons to arrange and out of 10, six people can sit on one side as only 6 seats will be reaming after making 2 people sit on one side on special demand and 4 people on other side as 4 people are already being seated on one side on special demand. (Take into consideration that one side has only 8 seats).

The number of ways of choosing 6 people out of 10 are ,

Now there are 4 people remaining and they will automatically place in those 4 seats available on the other side therefore total arrangement for those four people are

And now all the 16 people are placed in their seats according to the constraints.

Now we have to arrange them.

So, the number of ways of arranging 8 people out of 16 on one side and the rest 8 people on other side is(8!×8!).

So, a possible number of arrangements will be

Now as we know

So total number of arrangements is

= 210×(8! × 8!)

There are 16 people for the tea party.

People sit along a long table with 8 chairs on each side.

Out of 16, 4 people sit on a particular side and 2 sit on the other side.

Therefore first we will make sitting arrangements for those 6 persons who want to sit on some specific side.

Now we have remaining (16 - 6) =10 persons to arrange and out of 10, six people can sit on one side as only 6 seats will be reaming after making 2 people sit on one side on special demand and 4 people on other side as 4 people are already being seated on one side on special demand. (Take into consideration that one side has only 8 seats).

The number of ways of choosing 6 people out of 10 are ,

Now there are 4 people remaining and they will automatically place in those 4 seats available on the other side therefore total arrangement for those four people are

And now all the 16 people are placed in their seats according to the constraints.

Now we have to arrange them.

So, the number of ways of arranging 8 people out of 16 on one side and the rest 8 people on other side is(8!×8!).

So, a possible number of arrangements will be

Now as we know

So total number of arrangements is

= 210×(8! × 8!)

### 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 ?</span

Maths-General

DETAILED SOLUTION:

There are 16 people for the tea party.

People sit along a long table with 8 chairs on each side.

Out of 16, 4 people sit on a particular side and 2 sit on the other side.

Therefore first we will make sitting arrangements for those 6 persons who want to sit on some specific side.

Now we have remaining (16 - 6) =10 persons to arrange and out of 10, six people can sit on one side as only 6 seats will be reaming after making 2 people sit on one side on special demand and 4 people on other side as 4 people are already being seated on one side on special demand. (Take into consideration that one side has only 8 seats).

The number of ways of choosing 6 people out of 10 are ,

Now there are 4 people remaining and they will automatically place in those 4 seats available on the other side therefore total arrangement for those four people are

And now all the 16 people are placed in their seats according to the constraints.

Now we have to arrange them.

So, the number of ways of arranging 8 people out of 16 on one side and the rest 8 people on other side is(8!×8!).

So, a possible number of arrangements will be

Now as we know

So total number of arrangements is

= 210×(8! × 8!)

There are 16 people for the tea party.

People sit along a long table with 8 chairs on each side.

Out of 16, 4 people sit on a particular side and 2 sit on the other side.

Therefore first we will make sitting arrangements for those 6 persons who want to sit on some specific side.

Now we have remaining (16 - 6) =10 persons to arrange and out of 10, six people can sit on one side as only 6 seats will be reaming after making 2 people sit on one side on special demand and 4 people on other side as 4 people are already being seated on one side on special demand. (Take into consideration that one side has only 8 seats).

The number of ways of choosing 6 people out of 10 are ,

Now there are 4 people remaining and they will automatically place in those 4 seats available on the other side therefore total arrangement for those four people are

And now all the 16 people are placed in their seats according to the constraints.

Now we have to arrange them.

So, the number of ways of arranging 8 people out of 16 on one side and the rest 8 people on other side is(8!×8!).

So, a possible number of arrangements will be

Now as we know

So total number of arrangements is

= 210×(8! × 8!)

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