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Math 372
Sample Quiz # 3
FORMULAS:
1. The number of r-combinations of {∞.a1 , ∞.a2 , ∞.a1 , . . . ∞.an }
is equal to
the number of ways to distribute r identical objects to n different boxes
is equal to
the number of non-negative integral solutions of x1 + x2 + x3 + . . . + xn = r
is equal to
r+n−1
.
r
2. The number of permutations of {n1 .a1 , n2 .a2 , n3 .a3 , . . . nk .ak }
is equal to
(n1 +n2 +...+nk )!
n1 ! n2 ! ... nk ! .
3. The number of r-permutations of {∞.a1 , ∞.a2 , ∞.a3 , . . . ∞.an }
is equal to
the number of ways to distribute r distinct object to n different boxes
is equal to
nr .
EXERCISES
5.3 (Page 207)
5.4 (Page 216)
PROBLEMS
1-13, 15-23, 25, 26
1-27, 37
1. Find the number of permutations of {a, 3.b, 4.d, 2.e, f }.
2. How many ways are there to roll a die six times and obtain a sequence of outcomes with one 1,
three 5s, and two 6s?
3. How many ways are there to arrange the letters in MISSISSIPPI?
4. How many arrangements are there of 4n letters, four of each of n types of letters, in which each
letter is beside a similar letter?
5. How many numbers greater than 3,000,000 can be formed by arrangements of 1,2,2,4,6,6,6?
6. Find the number of 5-permutations of {∞.a, ∞.b, ∞.c, ∞.d}.
7. What fraction of the 6-digit numbers formed with the digits 3, 5, and 7 have two 3s, two 5s, and
two 7s?
8. Find the number of 5-permutations of {4.a, 3.b, 2.d, e, f }
9. Find the number of 10-combinations of {∞.a, ∞.b, ∞.c}.
10. Find the number of 10-combinations of {15.a, 15.b, 10.c, 10.d}.
11. Find the number of 10-combinations of {15.a, 15.b, 15.c, 15.d} with at least two as and two bs.
12. Find the number of 10-combinations of {15.a, 15.b, 15.c, d, e, f }.
13. If three idetical dice are rolled, how many different outcomes can be recorded?
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14. Ten different peoplewalkinto a delicatessen to buy a sandwich. Four always order tuna fish, two
always order chicken, two always order roast beef, and two order any of the three types of sandwiches.
(a) How many different sequences of sandwiches are possible?
(b) How many different (unordered) collections of sandwishes are possible?
15. How many 8-digit sequences are there involving exactly six different digits?
16. How many 9-digit numbers are there with twice as many different odd digits involved as different
even digits?
17. How many ways are there to distribute 6 distinct apples and 8 peras to 3 distinct people such that
each person has at least one pear?
18. How many ways are there to distribute 6 distinct apples to 3 distinct people such that each person
has at least one apple?
19. How many ways are there to pick a collection of 10 coins from piles of pennies, nickles, dimes and
quarters?
20. How many ways are there to pick a selection of coins from 100 pennies, 20 nickels, and 10 dimes,
if you select:
(a) 10 coins?
(b) 15 coins?
21. How many ways are there to place nine differen rings on the four different fingers of your right
hand if:
(a) The order of rings on a finger does not matter?
(a) The order of rings on a finger is considered?
22. How many ways are there to select an unordered group of 6 numbers between 1 and 15 inclusive
with repetition? In what fraction of these ways is the sum of these numbers odd?
23. How many ways are there to distribute 7 identical balls into 10 different boxes such that each box
has at most one ball?
24. How many ways are there to distribute 7 distinct balls into 10 different boxes such that each box
has at most one ball?
25. How many ways are there to distribute 10 distinct balls into 10 different boxes such that no box
is empty?
26. How many ways are there to distribute 10 distinct balls into 10 different boxes such that exactly
one box is empty?
27. How many ways are there to distribute 10 distinct balls into 10 different boxes such that exactly
two boxes are empty?
28. How many arrangements of six 0s, five 1s, and four 2s are there in which:
(a) The first 0 precedes the first 1?
(b) The first 0 precedes the first 1, which precedes the first 2?
29. How many arrangements of letters in REPETITION are there with the first E occurring before
the first T?
30. How many arrangements of the letters in MATHEMATICS are there in which TH appears together
but the TH is not immediately followed by an E?
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31. How many arrangements of the letters in PEPPERMILL are there with:
(a) The M appearing to the left of all the vowels?
(b) The first P appearing before the first L?
32. How many binary sequences of length 20 are there that
(a) Start with a run of 0s, then a run of 1s, then a run of 0s, then a run of 1s, and finall finishing
with a run of os?
(b) Repeat part (a) with the constraint that each run is of length at least 2.
33. How many binary sequences of length 18 are there that start with a run of 1s, then a run of 0s,
then a run of 1s, and then a run of 0s, and such that one run of 1s have length at least 8?
34. How many integer solutions are there to x1 + x2 + x3 + x4 + x5 = 28 with:
(a) xi ≥ 0
(b) xi > 0
(c) xi ≥ i (i=1,2,3,4,5)
35. How many integer solutions are there to x1 + x2 + x3 = 15 with xi ≥ −5?
36. How many positive integer solutions are there to x1 + x2 + x3 + x4 < 100 ?
37. How many numbers between 0 and 10,000 have a sum of digits:
(a) Equal to 7?
(b) Less than or equal to 5?
(c) Equal to 14?
38. How many arrangements of 5αs, 5βs, and 5γs are there with no pair of consecutive αs?
39. How many ways are there to arrange the letters in VISITING with no pair of consecutive Is?
40. How many ways are there to arrange the 26 letters of the alphabet so that no pairs of vowels
appear consecutively?
41. If you flip a coin 20 times and get 14 heads and 6 tails, what is the probability that there is no
pair of consecutive tails?
42. Suppose a coin is tossed 12 times and there are 3 heads and 9 tails. How many such sequences are
there in which there are at least 5 tails in a row?
43. What fraction of all arrangements of EFFLORESCENCE has consecutive Cs and consecutive Fs
but no consecuive Es?
44. How many arrangements of MISSISSIPPI are there with consecutive Ps but no pair of consecutive
Ss?
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