Question

In how many ways 3 friends Ram, Rajat and Rupesh having 6 one rupee coins, 7 one rupee coins, 8 one rupee coins respectively donate 10 rupee coin collectively?

• A. 40
• B. 45
• C. 47
• D. 50

Answer 1

The correct option is C

47

Number of ways they can donate Rs. 10 is

$x_1+x_2+x_3=\mathrm{10.....}\left(1\right)$
Total ways are $={}^{10+3-1}{C}_{3-1}=66$
now
$x_1\le 6,x_2\le 7$ and $x_3\le 8$
So from equation $\left(1\right)$ subtracting the following cases
case 1
$x_1>6$
case 2
$x_2>7$
case 3
$x_3>8$

now
when $x_1>6$
then
$x_1+x_2+x_3=3$
so total ways are $={}^{3+3-1}{C}_{3-1}=10$
when $x_2>7$
then
$x_1+x_2+x_3=2$
so total ways are $={}^{2+3-1}{C}_{3-1}=6$
when $x_3>8$
then
$x_1+x_2+x_3=1$
so total ways are $={}^{1+3-1}{C}_{3-1}=3$

required ways are $66-10-6-3=47$