Practice problems for Midterm #1
MATH 31011, Fall 2007
The exam covers: Parity (see the handout with problems), the irrationality of √2 and
infinitude of the primes (3.7), induction (4.2, 4.3, 4.4), and combinatorics (6.1 - 6.7).
Note: On the exam, you do not need a numerical answer unless you are asked for it.
It is better to not simplify. Expressions like a
b,P(n, k) and n! are encouraged in your
answers. No calculators will be allowed!
The actual exam will have fewer questions.
(1) Give a numerical value for 10
7.
(2) Give a numerical value for P(6,4).
(3) How many ways are there to place 20 identical balls into 3 distinct boxes?
(4) How many ways can the 25-member cooking club choose its president, vice presi-
dent, and secretary?
(5) How many ways can 22 players be divided into two teams of 11 for a soccer game?
(6) How many distinct arrangements of the letters in HIPPOPOTAMUS are there?
(7) How many numbers less than 800,000 can be formed by rearranging the digits in
219,338?
(8) How many ways can you arrange the letters of the English alphabet so that there
are exactly 5 letters between the aand the b?
(9) Count the number of 5-card poker hands with 4 of the same denomination.
(10) What is the probability that after a pair of dice is rolled the smallest of the two
numbers showing is 4?
(11) How many strings of 4 letters begin or end with one of the five vowels?
(12) How many integers from 1 to 300 are divisible either by 4 or by 14?
(13) Compute the sum
n
0−n
1+n
2−n
3· ·· + (−1)n−1n
n−1+ (−1)nn
n
(Hint: Use the Binomial Theorem.)
(14) Prove by induction that for any n≥1
1·21+ 2 ·22+ 3 ·23+· ·· +n·2n= 2 + (n−1)2n+1
(15) Prove by induction that (1 + x)n≥1 + nx for any positive integer nand any
x≥ −1.
(16) When a group of nbusinessmen arrives at a meeting each person shakes hands
with all the other people present. What is the number of handshakes?