In how many ways can 3 prizes be distributed among seema jyoti bhavna and niru

Uh-Oh! That’s all you get for now.

We would love to personalise your learning journey. Sign Up to explore more.

Sign Up or Login

Skip for now

Uh-Oh! That’s all you get for now.

We would love to personalise your learning journey. Sign Up to explore more.

Sign Up or Login

Skip for now

Answer

In how many ways can 3 prizes be distributed among seema jyoti bhavna and niru
Verified

Hint: a) We will choose 3 boys among the 4 to give prizes and multiply the permutation of the prizes within themselves as well. b) We need to fill in these blanks with 4 things with repetition being allowed. c) We will just subtract the possibilities of 1 boy getting all the prices from the number of ways we calculated in b).

Complete step-by-step answer:

A) We need to distribute 3 prizes to 4 boys and no boy gets more than 1 prize. So, we basically need to select 3 distinct boys for the 3 prizes.We need to choose 3 among 4 which can be done in $^4{C_3}$ ways.We know that $^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$.Hence, $^4{C_3} = \dfrac{{4!}}{{3!(1)!}} = \dfrac{{4 \times 3 \times 2 \times 1}}{{3 \times 2 \times 1}} = 4$ ways.And, corresponding to these 4 ways, the prizes can be permuted as well.Since there are 3 prizes, therefore, they can be permuted in 3! ways which is $3 \times 2 \times 1 = 6$ ways.Hence, the total number of ways 3 prizes can be distributed to 4 students when no boy gets more than 1 prize is $4 \times 6 = 24$ ways.B) We need to distribute 3 prizes to 4 boys when anyone can win any number of prizes. We can treat it as three blank spaces which need to be filled with 4 distinct things when repetition of things is allowed as well. So, we have “_ _ _” which can be done in $4 \times 4 \times 4 = 64$ ways.C) We need to distribute 3 prizes to 4 boys when one boy cannot win all the prizes. So, we basically need to subtract the possibilities of one boy getting all the prizes from the number of ways we calculated in (b).We see that if all the prizes go to one boy only. This can be done in 4 ways since there are 4 boys to distribute prizes to. Hence, we need to subtract 4 from 64.

Hence, the answer is 64 – 4 = 60 ways.

Note: The students must think clearly why did we subtract the 4 from 64 in the part c). We did this because in part c) we basically need to find the number of ways we can distribute 3 prizes among 4 boys, when any boy cannot get all the prizes and in part b) we actually find the total possibility of distributions possible because there was no restriction in b). Permutation and Combinations have made our life so easy that we can calculate the number of arrangements without actually getting into writing all the arrangements.