Math 1004: Midterm 1 Practice
Monday, September 26th
1. The alphabet has 21 consonants and 5 vowels. How many 4 letter words can be made
if...
(a) There are no restrictions on the letters?
(b) There are no repeated letters?
(c) The word starts with a vowel and ends with the letter P?
(d) The letters in the word are all consonants or all vowels?
(e) The letters in the word are all different and are in alphabetical order (e.g. CJRZ
or JNOR or ABCD)?
(f) There are three consonants?
(g) There is at least one repeated letter?
2. There are 10 family members, of which 5 are male.
(a) How many ways can at least two of them go to the park?
(b) How many ways can four of them go to the store and choose somebody to drive?
(c) How many ways can two males and two females go to the park?
(d) How many ways can the family sit around a round table?
3. You and a friend are going out for dinner. You can choose between two restaurants and
each restaurant has 4 meals. Assuming that you and your friend get different meals,
how many different ways can dinner happen?
4. How many 5 letter words can you make with exactly two repeated letters?
5. How many ways can the letters in the word “FINITE MATHEMATICS” be used to
make a two word phrase, where all the letters of “FINITE” are used for the first word
and all the letters of “MATHEMATICS” are used for the second?
6. What is the coefficient of x3 y 2 in (2x + y)5 ?
7. Let G be the set of people who can play guitar, D be the set of people who can play
drums, and S be the set of people who can sing. Describe the following sets:
(a) G ∩ D ∩ S
(b) G ∪ D0
(c) People who cannot sing or play guitar
(d) People who play drums and cannot sing
8. Let S and T be subsets of the universal set U. Sketch a Venn diagram for the data below
by determining the number of elements in each basic region.
n(U ) = 23, n(S) = 13, n(T ) = 12, n(S ∪ T ) = 17
Math 1004 (Finite Probability): Fall 2016
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9. Suppose you draw a 5 card hand from a standard 52 card deck.
(a) How many ways can you draw the 5 card hand?
(b) How many hands have exactly 3 spades?
(c) How many hands have exactly 2 kings?
(d) How many hands have at least one king?
(e) (Tricky) How many hands have exactly 2 kings or exactly 2 aces?
(f) How many hands are a flush (all the same suit)?
(g) How many hands have cards all of different ranks?
(h) How many hands are a full house (two of one rank and three of another)?
(i) (Tricky) How many hands are a two pair (exactly two cards of one rank, two of
another rank, and one of a third rank)?
Answers
1. (a) 264
(b) P (26, 4)
(c) 5 · 26 · 26 · 1
(d) 214 + 54
(e) C(26, 4)
(f) 4 · 5 · 213 (choose where the vowel goes, choose the vowel, and choose three consonants)
(g) 264 − P (26, 4) (total words minus number with no repeats)
2. (a) 210 − C(10, 0) − C(10, 1)
(b) C(10, 4) · 4
(c) C(5, 2) · C(5, 2)
Math 1004 (Finite Probability): Fall 2016
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(d)
10!
10
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= 9!
3. 2 · 4 · 3 = 24
(Choose four letters from the alphabet and choose one of those to get
4. C(26, 4) · 4 · 5!
2!
doubled, then arrange those 5 letters into a word keeping in mind that two letters are
identical)
5.
6!
2!
·
11!
2!2!2!
6. The term is
5
2
(2x)3 y 2 = 80x3 y 2 , so that the coefficient is 80.
7. (a) People who can play guitar and drums and can sing.
(b) People who can play guitar or who cannot play drums.
(c) (S ∪ G)0
(d) D ∩ S 0
8. Use the inclusion-exclusion principle to find n(S ∩ T ) = 8, from which it’s easy to then
fill in an appropriately labeled Venn diagram.
9. (a) C(52, 5)
(b) C(13, 3) · C(39, 2)
(c) C(4, 2) · C(48, 3)
(d) C(52, 5) − C(48, 5)
(e) C(4, 2) · C(48, 3) + C(4, 2) · C(48, 3) − C(4, 2) · C(4, 2) · 44 (number of hands with
two kings plus number of hands with two aces minus the number of hands that
have both two kings and two aces)
(f) 4 · C(13, 5)
(g) C(13, 5) · 45
(h) P (13, 2) · C(4, 3) · C(4, 2)
(i) C(13, 2) · C(4, 2) · C(4, 2) · 44