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### Question

#### How many signals can be given by means of 10 different flags when at a time 4 flags are used, one above the other?

^{10}P_{4}
^{10}C_{4}
- 4
- 40

^{10}P_{4}^{10}C_{4}### Hint:

**Use formula**

#### The correct answer is: ^{10}P_{4}

#### You are given 10 flags of different colors and you can give signals, but use only 4 flags.

First you have to choose 4 flags from 10 flags

This can be done in ways

Among this 4 flags of different colors, 4! ways of different arrangements are possible

So total number of signals =

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#### 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.

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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.

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#### 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

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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 ,

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

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

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

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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!).

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