Find the orders of all elements in Group D4({1,2,3,4,5,6}under x mod 7)

How many subsets of the set {1,2,3,4,5,6} have at least 3 elements?  The coefficient of x2y2 in the expansion of (2x + 3y)4 is

Recall that an affine cipher, fa,b(n) = an + b( mod 26), where a, b ∈ {0, 1, 2, . . . , 25}, is said to be valid if gcd(a, 26) = 1. (a)Use the key (a, b) = (3, 0) to come up with the cipher text corresponding to the following plaintext DO WELL.  (b) Now use the key (a, b) = (5, 1) to come up with the cipher text for DO WELL.  (c) The key (a, b) = (5, 1) was used to generate the following cipher text. Decrypt the message. gbidxv vbfev

Prove that among any group of people, there are always two people with the samenumber of friends within the group!

Suppose a permutation s swaps the first two elements of {1, 2, ..., n}. A permutation r rotates the elements: r(1)=2, r(2)=3, ... r(n-1)=n, r(n)=1. We can thus write in cycle notation: s=(1 2), r=(1 2 3 4 ... n). What does the permutation rm sr -m do?