Question
GeneralGeneralGeneral

For 3 ≤ r ≤ 30,

³⁰C30−r + 3(³⁰C31−r) + 3(³⁰C32−r) + ³⁰C33−r = mCr

Find m.

Verified Answer

Observe coefficients:

1, 3, 3, 1

These are coefficients in:

(1+x)³

Applying Vandermonde's identity:

ⁿCk + 3ⁿCk−1 + 3ⁿCk−2 + ⁿCk−3 = ⁿ⁺³Ck

Therefore:

m = 30 + 3

= 33

Final Answer

33