Bars and stars approach

#1

hi
i understood this approach, stars and bars but plz explain this case
let you have n=10
k=5
so first i filled first 5 options now i am left with k blanks with 5 options each
for every blank there is 5 options so answer should be 5^5

#2

You don’t have that many cases. Try for a smaller scenario, n=4,k=2.
So after assigning the ones to each, you are left with 2.
Now the possible ways according to you are 2^2.
But actually it is 3
2 0
0 2
1 1.
I don’t know if you remember but this problem is similar to find the non - negative integral solutions of an equation.
x1 + x2 + x3 + … + xk = n
Now because all xi >= 1
you subtract k from the LHS.
and the resulting solution is (n-k+k-1) C (k-1)

#3

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