Chapter 6: Counting sections 4-6

Matching Question Match each left-hand side of an identity involving binomial coefficients with its right-hand side. 1. ∑nk=0(-1)kC(n,k) = 2. ∑nk=0C(n,k) = 3. ∑nk=0(−2)kC(n, k) = 4. ∑nk=03kC(n,k) =

1. 0 2. 2^n 3. (-1)^n 4. 4^n

Matching Question Match each description of a number on the left with its value on the right.

1. 21 2. 60 3. 35 4. 63

1. There are ____________ways to select six bills from 10 $1 bills, eight $5 bills, 12 $10 bills, 25 $20 bills, and 18 $100 bills. (Assume that the order the bills are chosen does not matter and that the bills of each denomination are indistinguishable.) 2. There are ________________ways to select eight bills from 30 $1 bills, 20 $10 bills, 45 $20 bills, and 25 $100 bills. (Assume that the order the bills are chosen does not matter and that the bills of each denomination are indistinguishable.)

1. 210 2. 165

Ordering Question Click and drag on elements in order Put these statements in order to make a combinatorial proof of Pascal's identity, which says that if n and kare positive integers with n ≥ k, then C(n + 1, k) = C(n, k - 1) + C(n, k).

1. Suppose that T is a set containing n+1 elements. 2. Let a be an element of T, and let S=T-{a}. 3. There are C(n+1,k) subsets of T containing k elements. Each subset either contains a and k-1 elements of S or k elements S and not a. 4. There are C(n, k-1) subsets of k elements of T containing a and k-1 element of S, and C9n,k) subsets of k elements of T, all from S. 5. Therefore

Matching Question Match each description of a number on the left with its value on the right. 1. The number of r-permutations of n objects with no repetition allowed 2. The number of r-permutations of n objects with repetition allowed 3. The number of r-combinations of n objects with no repetition allowed 4. The number of r-combinations of n objects with repetition allowed

1. n! / (n-r)! 2. n^r 3. n! / r!(n-r)! 4. (n+r-1)! / r!(n-1)!

Matching Question Match each action on the left with the model on the right that facilitates finding the number of ways to perform the action. 1. Distribute hands of seven cards to each of three people. 2. Place 100 different employees into four unlabeled rooms. 3. Put 50 copies of the same toy into 10 unlabeled boxes, where each box can contain as many as 100 toys. 4. Put 10 unlabeled balls into eight labeled bins.

1.Distinguishable objects and distinguishable boxes 2.Distinguishable objects and indistinguishable boxes 3. Indistinguishable objects and indistinguishable boxes 4. Indistinguishable objects and distinguishable boxes

What is the next permutation in lexicographic order after 213465?

213546

How many different strings can be made by reordering the letters in the word OUAGADOUGOU?

277200

How many solutions does the equation x1 + x2 + x3+ x4 = 10 have in nonnegative integers?

286

What is the next permutation in lexicographic order after 321654?

324156

How many different strings can be made by reordering the letters of the word TENNESSEE?

3780

What is the next permutation in lexicographic order after 436125?

436152

How many solutions does the equation x1 + x2 + x3 + x4 + x5 = 8 have in nonnegative integers?

495

What is the next permutation in lexicographic order after 561234?

561243

How many solutions does the equation x1 + x2 + x3 + x4 + x5 + x6 = 7 have in nonnegative integers?

792

How many different strings can be made by reordering the letters in the word TALLAHASSEE?

831600

Complete the expansion of (2x - 3y)4 by filling in the coefficients. (2x - 3y)^4 = 16x^4 +_______x^3y + __________x^2 y^2 + _________xy^3 + __________y4

Blank 1. -96 Blank 2. 216 Blank 3. -216 Blank 4. 81

Use the algorithm for generating r-combinations to find the next larger 3-combination of the set {1, 2, 3, 4, 5} after {1, 3, 4}.

{1, 3, 5}

Use the algorithm for generating r-combinations to find the next larger 3-combination of the set {1, 2, 3, 4, 5} after {1, 4, 5}.

{2, 3, 4}

Use the algorithm for generating r-combinations to find the next larger 3-combination of the set {1, 2, 3, 4, 5} after {2, 3, 5}.

{2, 4, 5}

Use the algorithm for generating r-combinations to find the next larger 3-combination of the set {1, 2, 3, 4, 5} after {2, 4, 5}.

{3, 4, 5}