In how many different ways can three persons $A, B, C$ having 6,7 and 8 one rupee coins respectively, donate…
In how many different ways can three persons $A, B, C$ having 6,7 and 8 one rupee coins respectively, donate $₹ 10$ collectively?
$47$
$66$
$56$
$60$
Solution
Total number of ways $=[$ Number of ways in which $₹ 10$ can be distributed among three people]
-[Number of ways $A$ can have more than ₹ 6] -
[Number of ways $B$ can have more than $₹ 7$ ] -
[Number of ways $C$ can have more than ₹ 8]
Total possibilities
$={ }^{(n+r-1)} C_n={ }^{(10+3-1)} C_{10}={ }^{12} C_{10}=\frac{12 !}{10 ! 2 !}=66$
$A$ can have more than 6 in ${ }^5 C_3=10$ ways
$B$ can have more than 7 in ${ }^4 C_2=6$ ways
$C$ can have more than 8 in ${ }^3 C_1=3$ ways
$\therefore$ Total number of ways $=66-10-6-3=47$