CMSC 250 Homework 12 Fall 2005

0201 & 0202

Due Wed Nov 30 at the beginning of your discussion section.

Even though this is due after Thanksgiving Break, the material is on the second midterm exam.

You must write the solutions to the problems single-sided on your own lined paper, with all sheets stapled together, and with all answers written in sequential order or you will lose points. The answer only needs to be taken to a form that includes only addition, subtraction, multiplication, division, factorials, and exponents. You do not need to evaluate those to a number. Subparts of questions never affect the other subparts.

1. How many ways can 10 (identical) dimes be distributed among five children if
   1. there are no restrictions?
   2. each child gets at least one dime?
   3. the oldest child gets at least two dimes?
   4. the oldest child gets exactly two dimes?
2. A certain ice cream stor has 31 flavors of ice cream available. In how many ways can we order a dozen ice cream cones if
   1. we do not want the same flavor more than once?
   2. an individual flavor may be ordered as many as 12 times?
   3. an individual flavor may be ordered as many as 11 times?
3. Columba has two dozen each of n different colored beads. If she can select 20 beads (with repetition of color allowed) in 230,230 ways, what is the value of n?
4. Answer the following in the specifics of part a, and then generalize for part b.
   1. In how many ways can we select five coins from a collection of 10 consisting of one penny, one nickel, one dime, one quarter, one half-dollar and five (identical) Susan B. Anthony dollars?
   2. In how many ways can we select n objects from a collection of size 2n that consists of n distinct and n identical objects?
5. Determine the number of integer solutions of $x_1 + x_2 + x_3 + x_4 = 32$ where
   1. $x_i \geq 0$ for $1 \leq i \leq 4$
   2. $x_i > 0$ for $1 \leq i \leq 4$
   3. $x_1, x_2 \geq 5$ and $x_3, x_4 \geq 7$
   4. $x_1, x_2, x_3 > 0$ and $0 < x_4 \leq 25$

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