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?
Let x1,x2,x3,x4,x5,x6 be the number of apples adarsh can distribute.
The sum is less than or equal to 17.
Since adarsh can also keep some apples for himself introducing a dummy variable x7(x7≥0)
inequality becomes an equation
Here 1≤x1,x2,x3,x4,x5,x6≤6 and x7≥0
Number of solution = Coefficient of x17 in (x+x2+x3+.........x6)6(1+x+x2+x3+............)
Coefficient of x11 in (1−x6)6(1−x)−7
Coefficient of x11 in (1−6x6)(1−x)−7
Coefficient of x11 in (1−6x6)(1+7C1x+8C2x2+9C3x3+..............) = 17C11−611C6=17C6−611C5