Question

Two different families A and B are blessed with equal numbers of children. There are 3 tickets to be distributed amongst the children of these families so that no child gets more than one ticket.
If the probability that all the tickets go to the children of the family B is $\dfrac{1}{{12}}$, then the number of children in each family is?
A) 4
B) 5
C) 3
D) 6

Answer 1

Here, we will find the number of children in each family. We will find the number of ways that the tickets can be distributed among the children of the given families by using the combination. By using the probability, we will find the number of children in each by solving the linear equation.

Complete Step by step Solution:
We are given that two different families A and B are blessed with equal number of children.
Let $x$ be the children in both the families A and B.
We are given that gets more than one ticket and 3 tickets to be distributed amongst the children of these families.
So, we get
Total number of ways that the ticket is being distributed to two families $= {}^{2x}{C_3}$
Total number of ways that the ticket is being distributed to one of the two families B only $= {}^x{C_3}$
We are given that the probability that all the ticket go to the children of the family B is $\dfrac{1}{{12}}$.
So, we have
$\dfrac{{{}^x{C_3}}}{{{}^{2x}{C_3}}} = \dfrac{1}{{12}}$
By cross- multiplying, we get
$\Rightarrow 12{}^x{C_3} = {}^{2x}{C_3}$
$\Rightarrow \dfrac{{12x\left( {x - 1} \right)\left( {x - 2} \right)}}{{1 \cdot 2 \cdot 3}} = \dfrac{{2x\left( {2x - 1} \right)\left( {2x - 2} \right)}}{{1 \cdot 2 \cdot 3}}$
By cancelling the terms, we get
$\Rightarrow \dfrac{{12x\left( {x - 1} \right)\left( {x - 2} \right)}}{6} = \dfrac{{2x\left( {2x - 1} \right)\left( {2x - 2} \right)}}{6}$
$\Rightarrow 6x\left( {x - 1} \right)\left( {x - 2} \right) = 2x\left( {2x - 1} \right)\left( {x - 1} \right)$
$\Rightarrow 6\left( {x - 1} \right)\left( {x - 2} \right) = 2\left( {2x - 1} \right)\left( {x - 1} \right)$
By rewriting the equation, we get
$\Rightarrow 6\left( {x - 1} \right)\left( {x - 2} \right) - 2\left( {2x - 1} \right)\left( {x - 1} \right) = 0$
$\Rightarrow \left( {x - 1} \right)\left( {6x - 12 - 4x + 2} \right) = 0$
By simplifying the equation, we get
$\Rightarrow \left( {x - 1} \right)\left( {2x - 10} \right) = 0$
$\Rightarrow x - 1 = 0;2x - 10 = 0$
When $x - 1 = 0 \Rightarrow x = 1$ which is not possible to give 3 tickets for a child.
When $2x - 10 = 0 \Rightarrow x = \dfrac{{10}}{2} = 5$
Thus, the number of children is 5.

Therefore, the number of children in each family is 5.
Thus Option(B) is the correct answer.

Note:
We are using the concept of permutation. Permutation is defined as the arrangement of letters, numbers or some elements in a set. It gives us the number of ways that the elements in a set are arranged. Combination is defined as the selection of objects. Both are similar but in permutations order is important while in combinations order is not important. Factorial is defined as the numbers multiplied in the descending order till unity.