Download Document

Stats 241 Assignment 1
1. How many distinct ways can the letters of the word statistics be ordered.
Solution:
If the n = 10 letters of the word statistics were distinct there would be n! =
10! = 3,628,300 ways of ordering these letters. There are however 3 s’s, 3 t’s and two i’s.
For any ordering these three letters can be re-ordered 3!3!2! = 72 ways resulting in the
same ordering of the letters of the word statistics. Thus the total no. of distinct ways the
letters can be ordered is :
10!
3628200

 50400
3!3!2!
72
2. A three-digit numbers is formed from the digits 0, 1, 2, 3, 4, 5 with each digit used
only once? (a three-digit number does not start with 0)
a) How many such digits can be formed?
Solution:
Let N = the number of three-digit numbers that can be formed from the
digits 0, 1, 2, 3, 4, 5 if each digit can only be used once = N1N2N3 where
N1 is the number of ways of selecting the first digit = 5 (0 excluded)
N2 is the number of ways of selecting the 2nd digit = 5 (0 included, first digit selected
excluded)
N3 is the number of ways of selecting the 3rd digit = 4 (first two digits selected excluded)
Thus N = 5.5.4 =100
b) How many of these are odd numbers?
Solution:
To count the number of outcomes in this set, assign the 3rd digit first, the
st
1 digit second and the second digit third. Thus
N1 is the number of ways of selecting the last digit = 3 (from 1, 3 or 5)
N2 is the number of ways of selecting the 1st digit = 4 (0 and the last digit selected
excluded)
N3 is the number of ways of selecting the 2nd digit = 4 (first and last digit selected
excluded)
Thus N = 3.4.4 =48
c) How many are greater than 330?
Solution: Let A = {x | x  300 and is a number that satisfies the conditions of part
a)}and B = {x | 300 x  329 and is a number that satisfies the conditions of part a)}.
Let C = {x | x > 330 and is a number that satisfies the conditions of part a)}. Then
n(C) = n(A) – n(B) and
n(A) = 3.5.4 = 60 (since the first digit has to be chosen from {3,4,5}) and
n(B) = 3.4 =12 (since the first two digits has to be chosen from {30,31, 32}
Thus n(C) = 60 – 12 = 48
Page 1
Stats 241 Assignment 1
3. How many distinct ways can six trees be ordered when planted in a circle?
4. From a group of five men and three women a committee of three is selected at
random.
a) Find the probability that the committee contains two men and one woman.
b) Find the probability that the committee contains no men.
c) Find the probability that the committee contains no women.
5. How many integers are there between 1,000,000 and 10,000,000 in whose decimal
form no two consecutive digits are the same? (1,212,121 would be such an integer but
1,234,566 would not because the last two consecutive digits are the same.)
Solution:
999999999  99  387420489
Page 2
Stats 241 Assignment 1
6. What is the probability that a Bridge hand (13 cards) contains
a) 4 spades, 5 hearts, 3 diamonds and 1 club?
 13   13   13   13 
 4  5  3  1 
         9.45  1011
 52 
 13 
 
b) no spades?
 39 
 13 
   0.01279
 52 
 13 
 
c) a void? (at least one suit that is not represented)
Let S = the event that hand is void in spades.
Let H = the event that hand is void in hearts.
Let D = the event that hand is void in diamonds.
Let C = the event that hand is void in clubs.
P void   P S  H  D  C   P S   P  H   P  D  P C 
P S  H   P S  C   P S  D  P H  C   P H  D  P C  D
 P  S  H  D  P  S  H  C   P  S  C  D   P  H  D  C 
P S  H  D  C 
 39 
 26 
 13 
 13 
 13 
 13 




4
6
 4    0  0.051262
 52 
 52 
 52 
 13 
 13 
 13 
 
 
 
Page 3