Question

The number of ways in which $10$ identical apples can be distributed among $6$ children so that each child receives one atleast one apple is:

• A. $126$
• B. $252$
• C. $378$
• D. $noneofthese$

Answer 1

Answer: (A)

$126$

One child receive $5$ apples and the other $54$ children receives $1$ apple.There are$6$ ways to select a child who receives $5$ apples.

Hence, there are $6$$\cdot$$5$=$30$ such distribution.

Two children each receives three apples and the other four children each receive one apple

$\therefore$ ${}^6{C}_2=15$ ways.

One child receives $3$ apples, $24$ children each receive $2$ apples and the other $3$ children each receive one apple.There are $6$ ways to choose the child who receive $3$ apples and ${}^5{C}_2$ way to choose which $2$ of other$5$children each receive $2$ apples.

$⟹$ ${}^6{C}_1{\times }^5{C}_2=60$ distributions

$4$ children each receive two apples and the other two children receive one apple=${}^6{C}_4$=$15$ ways.

$\therefore$ Total $=6+30+15+60+15={}^9{C}_5=126$