Question

In a shop there are five types of ice creams available. A child buys six ice creams.
Statement 1: The number of different ways the child can buy the six ice-creams is
${}^{10}{C}_5$.
Statement 2: 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.

• A. Statement 1 is false, statement 2 is true.
• B. Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.
• C. Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.
• D. Statement 1 is true, statement 2 is false.

Answer 1

The correct option is A Statement 1 is false, statement 2 is true.

In shop there are five different types of ice creams. Let a child buy $x_i$ ice creams of type i, where i = 1, 2, 3, 4, 5.
Child buys six ice creams.
$\therefore x_1+x_2+x_3+x_4+x_5=6$ where $0\le x_i\le 6$
So total number of ways in which child can buy ice creams = number of non-negative solutions of above equations = placing 6 identical balls in 5 different boxes = ${}^{6+5-1}{C}_{5-1}{=}^{10}{C}_4$