A test has 30 multiple choice questions with 4 choices each. Which expressions represents the total number of ways to answer the questions?

Answer 1

Maybe apply Pascal's Triangle and the binomial expansion

If you guessed on 30 questions each with 4 choices the odds of all right would be (1/4)^30

the odds of all wrong would be (3/4)^30. odds of in between are given as

(30/30)1/4)^30 + (30/29)(1/4)^29(3/4) + (30/28)(1/4)^28(3/4)^2 + (30/27)(1/4)^27(3/4)^3 ....(30/n)(1/4)^n(3/4)^(30-n) ... (1/4)^30

the coefficients are the combinations of 30 taken n at a time = 30!/n!(30-n)! which are entries on a line from Pascal's triangle