Q172 / 270 · NDA 2022 (Phase 1) · Mathematics
What is the value of
$\begin{aligned} & 2(2 \times 1)+3(3 \times 2 \times 1)+4(4 \times 3 \times 2 \times 1)+ \\ & 5(5 \times 4 \times 3 \times 2 \times 1)+\ldots \ldots \ldots \ldots \ldots \ldots+ \\ & 9(9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)+2 ?\end{aligned}$
- A11 !
- Correct answerB10 !
- C$10+10$ !
- D$11+10$ !
Solution
$\begin{aligned} & 2(2 \times 1)+3(3 \times 2 \times 1)+4(4 \times 3 \times 2 \times 1)+5 \\ & (5 \times 4 \times 3 \times 2 \times 1)+\ldots \ldots . .+9(9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times \\ & 2 \times 1)+2 \\ & =2+2.2!+3.3!+4.4!+5.5!+\ldots .+9.9! \\ & =2!+(2+1)+3.3!+4.4!+5.5!+\ldots .+9.9! \\ & =3!+3.3!+4.4!+5.5!+\ldots .+9.9!\end{aligned}$
Proceed further we get
$=9!+9.9!=10!$
Hence, the correct answer is option: 'B'.