What is the remainder when 1! +2! +3! +4! ...+100! is divided by 24?

Answer = 9 All numbers from 4! and ahead are multiples of 24 (the value of 24 itself being 4! and 5! = 5*4! and similarly all n! are multiples of 24 $$\forall n \in N, n \ge 5 $$) So, take (mod24) on 1! + 2! + 3! ........... 100! - => 1!(mod24) + 2!(mod24) + 3!(mod24) + 4!(mod24) ............ 100!(mod24) => 1!(mod24) + 2!(mod24) + 3!(mod24) + 0(mod24) ..................0(mod24) {As 4! and further n! divide 24} => 1(mod24) + 2(mod24) + 6(mod24) = 9(mod24) => Remainder is 9

Find the derivative of $$f(x) = x.sin(x)$$ at $$x = \pi$$ using multiplication rule.

Multiplication rule - $$\frac{df(x)g(x)}{dx} = g(x).\frac{df(x)}{dx} + f(x).\frac{g(x)}{dx} $$ Here, $$f(x) = x$$ and $$g(x) = sin(x)$$. => $$\frac{d(xsin(x))}{dx} = sin(x).\frac{d(x)}{dx} + x.\frac{d(sin(x))}{dx} $$ => $$\frac{d(xsin(x))}{dx} = sin(x).+ x.cos(x) $$ => $$\frac{d(xsin(x))}{dx}_{x=\pi} = sin(\pi).+ \pi.cos(\pi) $$ => $$\frac{d(xsin(x))}{dx}_{x=\pi} = 0.+ \pi.(-1) = -\pi$$ => $$\frac{d(xsin(x))}{dx}_{x=\pi} = -\pi$$

There are 10 participants in a race. In how many different ways can the first, second, and third prize be awarded?

There are 3 slots available to fill- 1) 1st 2) 2nd 3) 3rd 1) There are 10 choices to choose the 1st place participant. 2) As there is a participant in 1st place, there are only 9 choices left for the second place. 3) Similarly, there are only 8 choices left for the third option. So, the total ways possible = (10 choices)*(9 choices)*(8 choices) = 720 ways to choose the prizes.