Maths
General
Easy
Question
The number of ways in which three persons can dress themselves when they have 4 shirts. 5 pants and 6 hats between them, is
 ^{4}C_{3} Ã— ^{5}C_{3} Ã—^{6}C_{3}
Â
Â
 ^{4}P_{3}Ã—^{ 5 }P_{3} Ã— ^{6 }P_{3}
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Hint:
Here, we can use the formula
The correct answer is: ^{4}P_{3}Ã—^{ 5 }P_{3} Ã— ^{6 }P_{3}
Given that,Â
Shirts = 4 ,Â Â Â Pants = 5Â andÂ Â Hats = 6, which are distributed among 3 men.
For shirts, Number of ways they can wear =Â
For pants, Number of ways they can wear =Â
For hats, number of ways they can wear =Â
Total number of ways they can wearÂ
Thus, the number of ways in which three persons can dress themselves when they have 4 shirts. 5 pants and 6 hats between them, isÂ .
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Eleven animals of a circus have to be placed in eleven cages, one in each cage. If four of the cages are too small for six of the animals, the number of ways of caging the animals is
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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
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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
Â 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 areÂ ,Â
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 areÂ
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!Ã—8!).
So, a possible number of arrangements will be
Now as we know
So total number of arrangements is
=Â 210Ã—(8!Â Ã—Â 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 areÂ ,Â
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 areÂ
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!Ã—8!).
So, a possible number of arrangements will be
Now as we know
So total number of arrangements is
=Â 210Ã—(8!Â Ã—Â 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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Â 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 areÂ ,Â
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 areÂ
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!Ã—8!).
So, a possible number of arrangements will be
Now as we know
So total number of arrangements is
=Â 210Ã—(8!Â Ã—Â 8!)
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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 areÂ ,Â
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 areÂ
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!Ã—8!).
So, a possible number of arrangements will be
Now as we know
So total number of arrangements is
=Â 210Ã—(8!Â Ã—Â 8!)
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