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# The number of numbers can be formed by taking any 2 digits from digits 6,7,8,9 and 3 digits from 1, 2, 3, 4, 5 is -

^{5}C_{3 }× ^{4}C_{2} × 3! × 2!
^{5}P_{3} × ^{4}P_{2} × 5!
^{5}C_{3} × ^{4}C_{2} × 5!
^{5}C_{3} × ^{4}C_{2} ×

^{5}C_{3 }×^{4}C_{2}× 3! × 2!^{5}P_{3}×^{4}P_{2}× 5!^{5}C_{3}×^{4}C_{2}× 5!^{5}C_{3}×^{4}C_{2}×Hint:

### No repetition is allowed.

Number of ways of taking any 2 digits from digits 6,7,8,9 = similarly find out the rest.

## The correct answer is: ^{5}C_{3} × ^{4}C_{2} × 5!

### We have to form a 5digit number by taking any 2 digits from digits 6,7,8,9 and 3 digits from 1, 2, 3, 4, 5

Number of ways of taking any 2 digits from digits 6,7,8,9 =

Number of ways of taking any 3 digits from 1, 2, 3, 4, 5 =

Number of ways of forming a 5 digit number = 5!

Thus, the number of numbers can be formed by taking any 2 digits from digits 6,7,8,9 and 3 digits from 1, 2, 3, 4, 5 is - 5! .

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### In how many ways can six different rings be wear in four fingers?

Detailed Solution

Now, we have been given that there are 6 rings of different types and we have to find the ways in which they can be worn in 4 fingers.

Now, we know that the number of options each ring has is 4, that is each ring has 4 fingers as their possible way as it can be worn in any one of 4 fingers.

Now, similarly the other rings will have four options as it has not been mentioned in the options that there has to be at least a ring in a finger. So, each ring has four options i.e. four fingers.

Now, we know that by the fundamental principle of counting there can be = 4 x 4× 4× 4× 4× 4 ways of wearing 6 rings.

So, we have = 4096 ways to wear 6 different types of rings.

Now, we have been given that there are 6 rings of different types and we have to find the ways in which they can be worn in 4 fingers.

Now, we know that the number of options each ring has is 4, that is each ring has 4 fingers as their possible way as it can be worn in any one of 4 fingers.

Now, similarly the other rings will have four options as it has not been mentioned in the options that there has to be at least a ring in a finger. So, each ring has four options i.e. four fingers.

Now, we know that by the fundamental principle of counting there can be = 4 x 4× 4× 4× 4× 4 ways of wearing 6 rings.

So, we have = 4096 ways to wear 6 different types of rings.

### In how many ways can six different rings be wear in four fingers?

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Detailed Solution

Now, we have been given that there are 6 rings of different types and we have to find the ways in which they can be worn in 4 fingers.

Now, we know that the number of options each ring has is 4, that is each ring has 4 fingers as their possible way as it can be worn in any one of 4 fingers.

Now, similarly the other rings will have four options as it has not been mentioned in the options that there has to be at least a ring in a finger. So, each ring has four options i.e. four fingers.

Now, we know that by the fundamental principle of counting there can be = 4 x 4× 4× 4× 4× 4 ways of wearing 6 rings.

So, we have = 4096 ways to wear 6 different types of rings.

Now, we have been given that there are 6 rings of different types and we have to find the ways in which they can be worn in 4 fingers.

Now, we know that the number of options each ring has is 4, that is each ring has 4 fingers as their possible way as it can be worn in any one of 4 fingers.

Now, similarly the other rings will have four options as it has not been mentioned in the options that there has to be at least a ring in a finger. So, each ring has four options i.e. four fingers.

Now, we know that by the fundamental principle of counting there can be = 4 x 4× 4× 4× 4× 4 ways of wearing 6 rings.

So, we have = 4096 ways to wear 6 different types of rings.

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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.

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

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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 ,

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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