Use Induction to prove the following.  You must clearly specify the base case and the inductive case. 11n - 6 is a multiple of 5 for all natural numbers n.

Use the pattern in Pascal's triangle to find the probability of getting 3 heads when five fair coins are tossed. The probability of getting 3 heads when five fair coins are tossed is

Use the formula for the sum of the first n integers to evaluate the sum given below, then write it in closed form. (For each answer, enter a mathematical expression.) (a) 6 + 7 + 8 + 9 + ⋯ + 600  (b) 6 + 7 + 8 + 9 + ⋯ + k

Find the formula for the nth term of the arithmetic sequ 3, 6, 9, ... an ||

Give an expression for the answer using permutation notation, combination notation, factorial notation, or other operations. Then evaluate. Of the first 11 questions on a test, a student must answer 7. Of the second 5 questions, the student must answer 3. In how many ways can this be done? Identify which notation properly depicts the evaluation to this problem. Select all that apply.

License plates in Oklahoma are of the form A BC123 4, where the first three spaces on the plate can be any letters from Á to Z and the last four spaces on the plate can be any numbers from 0 to 9. Assume that all possible combinations of letters and numbers are considered valid license plates. How many different plates are possible? A father wants the letters on his license plate to read "D A D." How many different plates could be made where the letters read “D A D"? A person witnessed a hit-and-run. They tell police that the license plate of the car that drove off was an Oklahoma license plate. The witness was sure that the first letter on the plate was an R, the first number on the plate was a 3, 6, or 8, and the last number on the plate was a 1 or a 7. How many different plates could exist with these conditions?

Six men and seven women enter a classroom. There are chairs numbered one through thirteen along the back wall. They must be seated at the chairs in such a way that the women are always in the odd numbered chairs and the men are always in the even numbered chairs. How many different seating arrangements are possible? (DO NOT INCLUDE COMMAS IN YOUR ANSWER)

A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes. Its base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions. 22 and 4 32 and 2 24 and 3 25 and 4

Please type the answer. I have already asked this question and couldn't read half the letters in the answer.

Discrete Math