Question

4 notes of Rs.100 and 5notes in which the first one is Rs.1, second of Rs.2, third of Rs.5, fourth of Rs.20 and fifth one of Rs.50 is distributed in three children such that each child receives at least one note of Rs.100. The total number of ways of distribution
a) $3\times {{5}^{3}}$
b) ${{5}^{3}}\times 3$
c) ${{3}^{6}}$
d) None of these

Answer 1

Hint:In this case, we have to find the number of ways in which the notes can be distributed. Now, it is given that each child should receive at least one note of Rs.100. Therefore, we should try to find out the total number of ways to distribute the remaining notes among the three children to obtain the answer to this question.

Complete step-by-step answer:
In this case, it is given that each child should receive at least one note of Rs.100. Therefore, we should try to calculate the possible ways of distributing the other notes among the three children.

When three 100-rupee notes are given to the three children, 6 notes remain. That they are 1note each of Rs.100, Rs.1, Rs.2, Rs.5, Rs.20 and Rs.50. Now, there is no restriction on the number of notes given to each child, therefore, once the three Rs.100 notes are distributed, each of the other notes can be given to any one of the children. Therefore, there are three ways to distribute each note…………………….. $(1.1)$
Therefore,
\[\begin{align}
  & \text{Total number of ways to distribute each note} \\
 & \text{= No}\text{. of ways to distribute the Rs}\text{.100 note }\!\!\times\!\!\text{ No}\text{. of ways to distribute the Rs}\text{.1 note} \\
 & \text{ }\!\!\times\!\!\text{ No}\text{. of ways to distribute the Rs}\text{.2 note} \\
 & \text{ }\!\!\times\!\!\text{ No}\text{. of ways to distribute the Rs}\text{.5 note }\!\!\times\!\!\text{ No}\text{. of ways to distribute the Rs}\text{.20 note} \\
 & \text{=3}\times \text{3}\times \text{3}\times \text{3}\times \text{3}\left( \text{Using equation 1}\text{.1} \right) \\
 & ={{3}^{6}} \\
\end{align}\]
Thus, the answer is ${{3}^{6}}$ which matches option (c). Therefore, option (c) is the correct answer to this question.

Note: As all the Rs.100 notes can be considered to be equivalent here, it does not matter which note is given to which child. Therefore, while calculating the number of ways of distributing the notes, we should consider the remaining one Rs.100 note along with the other notes in the calculation.