Question

The number of ways in which $30$ coins of one rupee each can be given to six persons so that none of them receives less than $4$ rupees is
A. $231$
B. $462$
C. $693$
D. $924$

Answer 1

In this question, we have to distribute $30$ coins of one rupee each to six persons in such a way that none of them receives less than $4$ rupees . All the coins are identical and have to be divided in “r” groups. The formula of distribution of such objects is $^{n + r - 1}{C_{r - 1}}$ , where $n$ is the objects and $r$ is the number of solutions which is required. We will use this formula to solve the above question.

Complete step by step answer:
In the question we have been given that none of them should receive less than $4$ rs. So let us first give $Rs\,4$ to each six people. It gives us total$Rs(6 \times 4) = Rs\,24$
Now $Rs\,24$ is distributed among six persons, then the number of coins that are remaining is: $30 - 24 = 6$
Now we have to distribute $6$ coins among six people. Here we have number of coins or the identical things that has to be distributed i.e.$n = 6$. And the number of solutions that we require or the number of persons is:
$r = 6$
By putting this in the formula we can write
$^{6 + 6 - 1}{C_{6 - 1}}$

On solving we have:
${ \Rightarrow ^{12 - 1}}{C_5}{ = ^{11}}{C_5}$
We will now find the value of this, by the formula of combination i.e.
$\dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
By comparing with the new equation we have
$n = 11,r = 5$
So we can write
$^{11}{C_5} = \dfrac{{11!}}{{5!\left( {11 - 5} \right)!}}$
On solving the values we have :
$\dfrac{{11 \times 10 \times 9 \times 8 \times 7 \times 6!}}{{5 \times 4 \times 3 \times 2 \times 1 \times 6!}} = \dfrac{{11 \times 10 \times 9 \times 8 \times 7}}{{5 \times 4 \times 3 \times 2 \times 1}}$
It gives the value
$\therefore 11 \times 6 \times 7 = 462$
So the required number of total ways is $462$.

Hence the correct option is B.

Note: In the above question we have applied two formulas of Combination.Combination is a way of selecting items from a collection, where the order of selection does not matter. The identical formula is used only when we have to distribute $n$ identical things among $r$ persons, where each one of them can receive $0,1,2$ or more items, even if some of them receive nothing. We should note that if we have to distribute $n$ identical things among $r$number of persons, but here they have to get something i.e. $1,2$ or more, then we use another formula i.e. $^{n - 1}{C_{r - 1}}$ .