Combinatorics Questions: Card Pairs, Letter Arrangements, and Number Sequences

Problem 1. Suppose we draw 9 cards from a standard 52-card deck. Find the probability that your 9-card hand consists of exactly 4 (distinct) pairs. Assume the cards are drawn at the same time (so order does not matter).  To receive full credit you must provide correct explanations in your solution.

 

Problem 2.  Consider the word

YABADABADOO

With sufficient explanation, find the number of ways to arrange it provided:

(i) [2 points] there are no restrictions.

(ii) [2 points] all the A's are consecutive.

(iii) [7 points] none of the A's are consecutive.

(iv) [9 points] neither BAD nor YOO appear.

 

Problem 3. Find the number of length-6 sequences (so order matters) from the numerical digits

{1,2,3,4,5,6,7,8,9}

such that the sequence uses exactly 3 different digits. To receive full credit you must provide correct explanations in your solution.