Question

Question 1 - Solve the following questions: 1.1 - Prove by the Principle of Mathematical Induction that 1 × 1! + 2 × 2! + 3 × 3! + ... + n × n! = (n + 1)! – 1 for all natural numbers n.

1.2 - a) How many license plates can be made using either four digits followed by five uppercase English letters or six uppercase English letters followed by three digits?

b) Seven women and nine men are on the faculty in the mathematics department. How many ways are there to select a committee of five members of the department if at most two women must be on the committee?

c) How many permutations of the letters ABCDEF contain the string CE?

Note: no hand writing answers pls.

Answer 1

Q1) The result is true for n =1 as and

Assume that the result is true for then

- - - - - - - - - (1)

Then for n=k we have

Hence the result is true for n =k as well

So by induction we conclude that