Question

In a shop, there are five types of ice-creams available. A child buys six ice creams.

Statement I. The number of different ways the child can buy the six ice creams is ${}^{10}C_5$.

Statement II. The number of different ways the child can buy the six ice creams is equal to the number of different ways of arranging $6$ $A\text{'}s$ and $4$ $B\text{'}s$ in a row.

• A. Statement I is correct, Statement II is correct; Statement II is a correct explanation for Statement I
• B. Statement I is correct, Statement II is correct; Statement II is not a correct explanation for Statement I
• C. Statement I is correct, Statement II is incorrect
• D. Statement I is incorrect, Statement II is correct

Answer 1

The correct option is D

Statement I is incorrect, Statement II is correct

Explanation for the correct statement:

Step 1. Check the first statement.

Let the five ice creams in the shop be $A,B,C,D,E,F$ and a children buys six ice creams, so number of ways in which he can buy the ice creams is the intergral solution of the equation $A+B+C+D+E+F=6$ and that is given as:

${}^{6+5-1}C_{5-1}={}^{10}C_4$

But it is given in statement I, that the number of ways is ${}^{10}C_5$. So the statement is incorrect.

Step 2. Check the second statement.

The number of different ways of arranging $6$ A’s and $4$ B’s in a row is given as:

$\begin{array}{rcl}\frac{10!}{6!4!}& =& \frac{10!}{4!(10-4)!}\\ & =& {}^{10}C_4\left[{}^{n}C_r=\frac{n!}{r!(n-r)!}\right]\end{array}$

So it is same as the number of ways in which the children can buy the ice creams.

So statement II is correct.

Thus, statement I is incorrect and statement II is correct.

Hence, the correct option is D.