Q1:

p = (1 × 1!) + (2 × 2!) + (3 × 3!) + (4 × 4!) + … + (10 × 10!)

Now, n × n! = [(n + 1) – 1] × n! = (n + 1)! – n!

∴ p = 2! – 1! + 3! – 2! + 4! – 3! + 5! – 4! +… + 11! – 10!

∴ p = 11! – 1! = 11! – 1

∴ p + 2 = 11! + 1

∴ p + 2 when divided by 11! leaves a remainder of 1.

Q2:

The points satisfying the equations x + y < 41, y > 0, x > 0 lie inside the triangle. Integer solutions of x + y < 41 can be found as follows. If x + y = 40

(x, y) (1, 39), (2, 38), …, (39, 1) … (39 solutions)

If x + y = 39 (1, 38), (2, 37), …, (38, 1) … ( 38 solutions)

If x + y = 38, we get 37 solutions and so on till x + y = 2 … (1 solution)

Thus there are 39 × 40/2 = 780 integer solutions to x + y < 41 The number of points with integer coordinates lying inside the circle = 780 Hence, option 1.