The sum of series 2C1 ⋅ (1 × 2) + 3C2 ⋅ (2 × 3) + 4C3 ⋅ (3 × 4) + ... + 19C18 is S,

Question

The sum of series \({ }^{2} C_{1} \cdot(1 \times 2)+{ }^{3} C_{2} \cdot(2 \times 3)+{ }^{4} C_{3} \cdot(3 \times 4)+\ldots+{ }^{19} C_{18} \cdot(18 \times 19)\) is S, then \(\frac{S}{295}\) is equal to

(1) 104

(2) 107

(3) 103

(4) 114 

Answer 1

Correct option is: (4) 114  

\(=\frac{18 \times 19 \times 20 \times 21}{4}-\frac{18 \times 19 \times 20}{3}\)

\(=18 \times 19\left[105-\frac{20}{3}\right]==\frac{18 \times 19}{3}[295]=33630\)

\(\frac{S}{295}=6 \times 19=114\)