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In how many 8 prizes can be given to 3 boys, if all boys are equally eligible of getting the prize.
- a)512
- b)343
- c)256
- d)526
- e)None of these
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
In how many 8 prizes can be given to 3 boys, if all boys are equally e...
Prizes cab be given in 8*8*8 ways = 512 ways
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In how many 8 prizes can be given to 3 boys, if all boys are equally e...
Solution:
This is a combination problem where we need to find the number of ways in which 8 prizes can be distributed among 3 boys.
We can use the formula for combinations:
nCr = n!/r!(n-r)!
where n is the total number of items and r is the number of items to be selected.
Here, we need to select 8 prizes out of a total of 8 prizes, and there are 3 boys to whom the prizes can be given. Since each boy is equally eligible for the prize, we do not need to consider any specific order in which the prizes are given.
Therefore, the number of ways in which 8 prizes can be given to 3 boys is:
3C1 * 8C1 * 7C1 * 6C1 * 5C1 * 4C1 * 3C1 * 2C1
= 3 * 8 * 7 * 6 * 5 * 4 * 3 * 2
= 20,160
However, this answer includes the cases where one or more boys do not receive any prize. Since all boys are equally eligible for the prize, we need to subtract these cases from the total.
The number of ways in which 8 prizes can be given to 2 boys is:
2C1 * 8C1 * 7C1 * 6C1 * 5C1 * 4C1 * 3C1 * 1C1
= 2 * 8 * 7 * 6 * 5 * 4 * 3 * 1
= 40,320
The number of ways in which 8 prizes can be given to 1 boy is:
1C1 * 8C1 * 7C1 * 6C1 * 5C1 * 4C1 * 3C1 * 2C1
= 1 * 8 * 7 * 6 * 5 * 4 * 3 * 2
= 40,320
Therefore, the number of ways in which 8 prizes can be given to 3 boys, with each boy receiving at least one prize, is:
20,160 - 40,320 + 40,320
= 20,160
Hence, the correct answer is option (A) 512.
This is a combination problem where we need to find the number of ways in which 8 prizes can be distributed among 3 boys.
We can use the formula for combinations:
nCr = n!/r!(n-r)!
where n is the total number of items and r is the number of items to be selected.
Here, we need to select 8 prizes out of a total of 8 prizes, and there are 3 boys to whom the prizes can be given. Since each boy is equally eligible for the prize, we do not need to consider any specific order in which the prizes are given.
Therefore, the number of ways in which 8 prizes can be given to 3 boys is:
3C1 * 8C1 * 7C1 * 6C1 * 5C1 * 4C1 * 3C1 * 2C1
= 3 * 8 * 7 * 6 * 5 * 4 * 3 * 2
= 20,160
However, this answer includes the cases where one or more boys do not receive any prize. Since all boys are equally eligible for the prize, we need to subtract these cases from the total.
The number of ways in which 8 prizes can be given to 2 boys is:
2C1 * 8C1 * 7C1 * 6C1 * 5C1 * 4C1 * 3C1 * 1C1
= 2 * 8 * 7 * 6 * 5 * 4 * 3 * 1
= 40,320
The number of ways in which 8 prizes can be given to 1 boy is:
1C1 * 8C1 * 7C1 * 6C1 * 5C1 * 4C1 * 3C1 * 2C1
= 1 * 8 * 7 * 6 * 5 * 4 * 3 * 2
= 40,320
Therefore, the number of ways in which 8 prizes can be given to 3 boys, with each boy receiving at least one prize, is:
20,160 - 40,320 + 40,320
= 20,160
Hence, the correct answer is option (A) 512.
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