15.13. A Magic Trick 501
(a) How many solutions over the natural numbers are there to the inequality
x 1 Cx 2 CCxnm?
(b) How many lengthmwords can be formed from ann-letter alphabet, if no
letter is used more than once?
(c) How many lengthmwords can be formed from ann-letter alphabet, if
letters can be reused?
(d) How many binary relations are there from setAto setBwhenjAj Dm
andjBjDn?
(e) How many injections are there from setAto setB, wherejAj Dmand
jBjDnm?
(f) How many ways are there to place a total ofmdistinguishable balls inton
distinguishable urns, with some urns possibly empty or with several balls?
(g) How many ways are there to place a total ofmindistinguishable balls inton
distinguishable urns, with some urns possibly empty or with several balls?
(h) How many ways are there to put a total ofmdistinguishable balls inton
distinguishable urns with at most one ball in each urn?Problem 15.25. (a)How many solutions over thepositiveintegers are there to the
inequality:
x 1 Cx 2 C:::Cx 10 100(b)In how many ways can Mr. and Mrs. Grumperson distribute 13 identical
pieces of coal to their three children for Christmas so that each child gets at least
one piece?
Problems for Section 15.10
Practice Problems
Problem 15.26.
The working days in the next year can be numbered 1, 2, 3,... , 300. I’d like to
avoid as many as possible.
On even-numbered days, I’ll say I’m sick. On days that are a multiple of 3, I’ll say I was stuck in traffic.