1) The number 123456789(10)(11)(12)(13)(14) is written in the base 15. If convert the number to base 10, it is equal to 1 × 1513 + 2 × 1512 + 3 × 1511 + ….+ 13 × 15 + 14 What is the the remainder upon dividing the base 10 number by 7?

How am I able to find out the remainder when the number is so big?? Please explain the working so that I understand it

2) Given that A = {a: a is integers}and B = {b: b is integers}where

1/a + 1/b = 1/14

Evaluate n(A × B).

how do I start the calculation so that i am able to find out the total number of choice for a and b??thanks

4) A pack of 8 distinct cards is shuffled randomly. What is the probability that at least one of the cards remains at the same place after the shuffle?

5) How many 10-digit quaternary (base 4) numbers (leading zeros are allowed) consist of

(a) exactly four 1s?

(b) exactly two 2s and three 3s?

(c) at least two 0s?

6) From where he stands, one step toward the cliff would send a drunken man over the edge. He takes random steps, either toward or away from the cliff. At any step his probability of taking a step away is 0.75, of a step toward the cliff is 0.25. What is his probability of escaping the cliff?

7) In a chess tournament, a contestant received 1 point for a win, 0.5 point for a tie. Two Accounting students entered a chess tournament otherwise composed of Computing students. Each contestant played once against each other contestant. The two Accounting students obtained a total of 8 points, and each Computing students scored the same number of points as his coursemates. How many Computing students participated in the chess tournament?

please explain how do you guys got the answer thanks!