Question

In a shop there are five types of ice-creams available. A child buys six ice-creams.
Consider the following statements
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's and 4 B's in a row.
Choose the correct option.

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

Answer 1

Answer: (D)

I. The number of ways the child can buy ice-cream $=$ The number of integral solutions
of $x_1+x_2+x_3+x_4 +x_5=6$
$={ }^{6+5-1} C_{5-1} $
$={ }^{10} C_4$
(number of integral solution of $x_1+x_2+\ldots .+x_r=n$ is $\left.{ }^{n+r-1} C_{r-1}\right)$
II. Number of ways of arranging 6 A's and 4 B's in a row
$=\frac{10 !}{6 ! 4 !}={ }^{10} C_4$