Question

find the next number 10, 45, 120, 210 ?

Answer 1

Start with 10

Multiply 10 by 9, get 90, divide by 2, get 45.

Multiply 45 by 8, get 360, divide by 3, get 120.

Multiply 120 by 7, get 840, divide by 4, get 210.

Multiply 210 by 6, get 1260, divide by 5, get 252. <--- answer

Answer: 252

Or you can do it this way with combinations of 10

things taken 1,2,3,4,5 at a time.

$^{10}C_1 = \frac{10!}{1! \times 9!} = \frac{10 \times 9!}{1 \times 9!} = \bold{10} \\ \\ ^{10}C_2 = \frac{10!}{2! \times 8!} = \frac{10 \times 9 \times 8!}{2 \times 8!} = \frac{90}{2} = \bold{45} \\ \\ ^{10}C_3 = \frac{10!}{3! \times 7!} = \frac{10 \times 9 \times 8 \times 7!}{6 \times 7!} = \frac{720}{6} = \bold{120} \\ \\ ^{10}C_4 = \frac{10!}{4! \times 6!} = \frac{10 \times 9 \times 8 \times 7 \times 6!}{24 \times 6!} = \frac{5040}{24} = \bold{210}$

$\therefore\ ^{10}C_5 = \frac{10!}{5! \times 5!} = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5!}{120 \times 5!} = \frac{30240}{120} = \bold{252}$

The sequence continues as,

10, 45, 120, 210, 252, 210, 120, 45, 10,......

Thank you.

#adithyasajeevan