of Review Chapter 5 Review
The Addition Principle The
r1 different objects in the first set, r2 different objects in the second set, disjoint , rm different objects in the mth set, # of ways to select an object form one of the ++ m sets: r1 + r2 ++ rm.
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The Multiplication Principle The
A procedure -> m successive (ordered) stages:
r1 different outcomes in the first stage, r2 different outcomes in the second stage, Independent distinct , rm different outcomes in the mth stage.
# of different composite outcomes the total procedure: r1 r2 rm.
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Summary Summary
Arrangement (ordered outcome) or Distribution of distinct objects No Repetition Unlimited Repetition Restricted Repetition Selection (unordered outcome) or Distribution of identical objects
P(n,r) nr P(r; r1, r2, , rn)
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C(n,r) C(r+n-1, r) ---4
Basic Formulas Basic
n! P ( n, k ) = k!
n n! C ( n, k ) = = k k!(n k )!
r! P(r ; r1 , r2 ,..., rn ) = r1!r2 !...rn !
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Integer-solution-of-an-equation version Integer-solution-of-an1. The number of ways to select r objects with repetition from n different types of objects. 2. The number of ways to distribute r identical objects into n distinct boxes. 3. The number of nonnegative integer solutions to x1+x2++xn=r.
12 =
4
+
3
+
1
+
4
6
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Probability Probability
Probability = # desired outcomes # total outcomes
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5.2#16 5.2#16
What is the probability that a five-card poker hand has the following? (b) four of a kind (9 9 9 9 J) (c) Two pairs (J J 4 4 9) (d) A full house (3 3 3 6 6) Sol: hw3
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Sample #2 Sample
What is the probability that a 7-card poker hand chosen from the 52 cards in a deck has exactly 3 pairs (no 3-of-a-kind or 4-of-a-kind)? Sol: sample
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Sample #5 Sample
How many ways are there to form a sequence of 8 letters from 4A's, 4B's and 4C's if each letter must appear at least twice? Sol: sample
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5.3 #22 5.3
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?
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5.3 #22 (a) sol 5.3
Find the positions for the subset (M,E,E,I) (C(10; 4) ways), order them and put the M in the first position of the subset (3 ways: MEEI, MIEE, MEIE). Then order the remaining 6 letters: 3P, 2L, 1R (P(6;3,2,1) ways). In total, there are C(10, 4)*3*P(6;3, 2, 1) = 37,800 ways.
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5.3 #22 (b) sol 5.3
Find the positions for the subset (P,P,P,L,L) (C(10,5)ways), order them and put a P in the first position of the subset (2P and 2L in the remaining 4 positions of the subset, so P(4;2,2)=6 ways). Then order the remaining 5 letters: 2E, 1M,1I and 1R (P(5; 2,1,1,1) = 20). In total, there are C(10,5)*6*20 = 30,240 ways.
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5.3 #26 5.3
How many ways are there to place nine different rings on the four fingers of your right hand (excluding the thumb) if
(a) The order of rings in a finger does not matter
Sol: distributing 9 distinct rings to 4 fingers, unlimited repetition, so the answer is 49.
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5.4 5.4 #7
How many ways are there to arrange the letters in VISITING with no pair of consecutive Is?
Sol: 3I, 1V, 1S, 1T, 1N, 1G. First order the 5 non-I letters: 5!. Then insert the 3Is to 6 possible positions: C(6,3). In total, there are 5!*C(6,3)=2400 ways.
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5.2 #45 5.2
What fraction of all arrangements of INSTRUCTOR has: (a) Three consecutive vowels?
Sol: 1N, 1I, 1S, 2T, 2R,1U, 1C, 1O. Glue the IOU together as one composite letter V (3! ways to order the subset). Then order the current 10-3+1=8 letters: 1N, 1S, 2T, 2R, 1C, 1V (P(8;1,1,2,2,1,1,) ways). In total, there are 3!*P(8;1,1,2,2,1,1)= 60,480 ways.
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Sample #3 Sample
How many arrangements of MATHEMATICAL contain IE, but do NOT contain IEA? Sol: sample
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Example Example
Consider the problem of counting the ways to select 25 objects from 8 types with at least 2 objects of each type. (a) Model this problem as an integer-solutionof-equation problem. sol: x1 + x2 + + x8 = 25, xi 2
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Three conditions 5.4 #47 Three
Q:How many arrangements of the letters in INSTITUTIONAL have all of the following properties simultaneously?
a) No consecutive Ts; b) The 2 Ns are consecutive; c) Vowels in alphabetical order.
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5.4 #7 sol 5.4
3T+( {2N, 1S, 1L},{3I, 1U, 1A, 1O})
Glue 2Ns together as one composite N. In the subset {1N, 1S, 1L},{3I, 1U, 1A, 1O}, since vowels are in alphabetical order, so fix their positions (C(9,6) ways), then order the remaining 3 letters: {1N, 1S, 1L} (3! ways). In the end, insert 3Ts to 10 possible positions (C(10,3) ways). In total, there are C(9,6)*3!*C(10,3) ways.
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Note Note
I made a mistake in class. I used C(12,6) instead of C(9,6) to choose the positions for the vowels. Actually in the 2nd step I only do with the subset of 9 letters. You can choose the positions for 3Ts first. When you have 3Ts+ 9 other letters (one is the composite N), you know 3Ts should be separate. So there are 10 positions to insert 3Ts: C(10, 3). Then you do the second step. Or you can treat the possibilities to insert letters to 3Ts as 4 boxes: ^T^T^T^, you need distribute the other 9 letters into those boxes so that at least 1 letter in the 2nd and 3rd boxes. This distribution question gives the same answer.
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Sample #7 Sample
How many arrangements of the letters in STATISTICAL have ALL of the following properties: 1. At least two consonants between successive vowels; 2. The arrangement starts with a vowel; 3. The consonants are NOT in alphabetical order. Sol: sample
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Review of Chapter 6 Review
6.1 6.1 #8
a) Use a generating function for modeling the number of different election outcomes in an election for class president if 27 students are voting among four candidates. Which coefficient do we want? b) Suppose each student who is a candidate votes for herself or himself. Now what is the generating function and the required coefficient? This is a hw question. Notice that we assume each student votes for one and only one candidate.
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Sample #1 Sample
Consider the problem of counting the ways to distribute 29 identical objects into 6 boxes with at least 4 objects in each box.
a) Model this problem as an integer-solution-of-anequation problem. b) Model this problem as a certain coefficient of a generating function. c) Solve this problem.
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Sample #4 Sample
Build a generating function for the following problems (You do NOT need to calculate the coefficient):
a) Select r balls from a pile of 3 red balls, 4 black balls, and 4 white balls. b) The number of ways there are to get a sum of 17 when 10 distinct dice are rolled.
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Sample #6 Sample
Find the corresponding coefficients:
1. What is the coefficient of x19 in the expansion of (x2 + x 3 + x 4 + )4 2. What is the coefficient of x10 in the expansion of (x+x2+x3)4(1+x+x2+x3+ )2?
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