p n c

is the ans to the first ques

¹⁰¹C₂ - 3⁵²C₂ = 1275

3⁵²C₂ .. What for is this part? can you explain your solution please.

"ways to choose 3 balls of all
minus ways not to choose any black ball "

i.e., 9C3-6C3 = 64 (Balls of same color are different)

I don't take credits

For the 1st question,

Let the shares to each son be S₁, S₂, S₃.

S₁ + S₂ + S₃ = 101

No. of integral solutions=¹⁰⁰C₂

Now, to remove those cases where one son gets more shares than the combined total of other two,

S₁ + S₂ ≤ 50

or,

S₁ + S₂ + x = 50

x ≥ 0

S₁ = 1+ a [a≥0]

S₂ = 1+ b [b≥0]

therefore,

a + b + x = 48

No . of integral solutions = ⁵⁰C₂

There will be three such cases.

So, no of ways to divide the shares = ¹⁰⁰C₂ - 3( ⁵⁰C₂)

Thanks!!