**P****lease ,****Solve the following questions **

**Q1 **

Prove by the Principle of Mathematical Induction that

1 × 1! + 2 × 2! + 3 × 3! + ... + *n *× *n*! = (*n *+ 1)! - 1 for all natural numbers *n*.

**Q2 **** **

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**?

**Q3**

**Q4**

4. What is the solution of the recurrence relation $a=−an−1)+4an−2)+4an−3)$

With $a=8,a=6,a=26?$

**Q5**

**Q6**

Find the first 3 terms in the expansion of $(3x−y)$ .