Question

How many ways can Adarsh distribute 17 identical apples among 6 students if he can also keep some apples for himself and each student should get at least one apple and not more than 6?

• A. -
• B. -
• C. -
• D. -

Answer 1

The correct option is D

-

Let $x_1,x_2,x_3,x_4,x_5,x_6$ be the number of apples adarsh can distribute.

The sum is less than or equal to 17.

$x_1+x_2+x_3+x_4+x_5+x_6$ $\le 17$

Since adarsh can also keep some apples for himself introducing a dummy variable $x_7(x_7\ge 0)$

inequality becomes an equation

$x_1+x_2+x_3+x_4+x_5+x_6+x_7=17$

Here $1\le x_1,x_2,x_3,x_4,x_5,x_6\le 6$ and $x_7\ge 0$

Number of solution = Coefficient of $x^{17}$ in $(x+x^2+x^3+.........x^6{)}^6(1+x+x^2+x^3+............)$

Coefficient of $x^{11}$ in $(1-x^6{)}^6(1-x{)}^{-7}$

Coefficient of $x^{11}$ in $(1-6x^6)(1-x{)}^{-7}$

Coefficient of $x^{11}$ in $(1-6x^6)(1{+}^7{C}_{1}x{+}^8{C}_2{x}^2{+}^9{C}_3{x}^3+..............)$ = ${}^{17}{C}_{11}-6^{11}{C}_6{=}^{17}{C}_6-6^{11}{C}_5$