NIMCET 2020 — Mathematics PYQ

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Question

NIMCET 2020 — Mathematics PYQ

NIMCET | Mathematics | 2020

There is a young boy's birthday party which 3 friends have attended. The mother has arranged 10 games where a prizeis awarded for winning a game. The prizes are identical. If each of the 4 children receives at least one prize, then howmany distributions of prizes are possible?

Choose the correct answer:

Explanation

Concept:
- The total number of ways in which n objects can be distributed in r people if any person can get 'any number of objects' = $n^{r - 1} C_{r - 1}$ .
- The total number of ways in which n objects can be distributed in r people if every person must get 'at least one object' = $n - 1 C_{r - 1}$ .
- $n C r = \frac{n !}{r ! ( n - r )!}$ .
- $n! = 1 \times 2 \times 3 \times ... \times n$ .
- $0! = 1$ .

Calculations:
Here, the number of objects is n = 10 and the number of people is r = 4.
Since, every person must get at least one object, the total number of possible distributions is:
$n - 1 C_{r - 1} = 10 - 1 C_{4 - 1} = 9 C_{3} = \frac{9 !}{3 ! ( 9 - 3 )!} = \frac{7 \times 8 \times 9}{3 \times 2 \times 1} = 84$ .