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 C5. 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.

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 C5. 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

Tardigrade Admin · Accepted Answer

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} + \dots . + x_{r} = n

is

^{n + r - 1} C_{r - 1})

II. Number of ways of arranging 6 A's and 4 B's in a row

= \frac{10 !}{6 ! 4 !} =^{10} C_{4}