Show your thinking before showing the numbers! You must show your
work completely and neatly for full credit.
1. I am adding to my standard 52-card deck two wild cards. (A "wild card"
is a playing card that can have any value, suit, color, or other property in
a game at the discretion of the player.) With this 54-card deck, how many
5-card hands give a "straight" (five cards with consecutive faces. The ace
can be used either as the lowest or the highest face)?
2. In a department of Mathematics with 20 faculty and 8 admin members,
a committee of 7 faculty and 3 admin must be formed. Also, two of the 7
chosen faculty members must be designated as chair and vice-chair of the
committee. In how many ways can this be done?
3. An urn contains 12 marbles: 3 blue, 4 red, 5 green. In how many ways
(a) line up all of them;
(b) line up five of them;
(c) extract five of them at once, without order.
4. Give a combinatorial proof of the following formula:
n b, C ) = 3".
(do not use the the Trinomial Formula!)
These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.