>>>Assuming that the 26 letters in the English language alphabet comprise a
population,

population = {A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z}

>>>1. Briefly explain how a simple random sample of size n = 7 can be obtained
with no mode(s). No calculations would be involved.

Get 26 identical slips of paper.  
Write a different letter on each one.
Put them in a hat.
Stir them up real good.
Draw out a slip of paper without looking.
Write down the letter it has written on it on a separate piece of paper.
Don't put it back (To make sure there is no mode.)
Keep drawing without putting any back until you have written down 7 letters.

>>>2. Showing your work, find the total number of simple random samples of size
n = 7 possible, none having any mode.

C(26,n) = [(26)(25)(24)(23)(22)(21)(20)]/[(7)(6)(5)(4)(3)(2)(1) = 657800

>>>3. Showing your work, what is the probability of drawing a simple random
sample of size n = 7 containing the set A, B, C, D, X, Y, Z, using a 6-ball
capacity scoop and at first attempt, from a bag containing 26 ping pong balls,
each uniquely labeled one of the 26 letters?

1/657800

Edwin