Introductory Combinatorics
5th Edition
ISBN: 9780134689616
Author: Brualdi, Richard A.
Publisher: Pearson,
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Chapter 5, Problem 10E
To determine
To prove: The identity
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7. The demand for a product, in dollars, is
p = D(x) = 1000 -0.5 -0.0002x²
1
Find the consumer surplus when the sales level is 200.
[Hints: Let pm be the market price when xm units of product are sold. Then the consumer
surplus can be calculated by foam (D(x) — pm) dx]
2. Claim events on a portfolio of insurance policies follow a Poisson process with parameter
A. Individual claim amounts follow a distribution X with density:
f(x)=0.0122re001, g>0.
The insurance company calculates premiums using a premium loading of 45%.
(a) Derive the moment generating function Mx(t).
4. Find the general solution and the definite solution for the following differential equations:
(a)
+10y=15, y(0) = 0;
(b) 2 + 4y = 6, y(0) =
Chapter 5 Solutions
Introductory Combinatorics
Ch. 5 - Prob. 1ECh. 5 - Fill in the rows of Pascal’s triangle...Ch. 5 - Consider the sum of the binomial coefficients...Ch. 5 - Expand (x + y)5 and (x + y)6 using the binomial...Ch. 5 - Expand (2x − y)7 using the binomial theorem.
Ch. 5 - What is the coefficient of x5y13 in the expansion...Ch. 5 - Use the binomial theorem to prove that
Generalize...Ch. 5 - Use the binomial theorem to prove that
Ch. 5 - Evaluate the sum
Ch. 5 - Use combinatorial reasoning to prove the identity...
Ch. 5 - Use combinatorial reasoning to prove the identity...Ch. 5 - Let n be a positive integer. Prove that
(Hint:...Ch. 5 - Find one binomial coefficient equal to the...Ch. 5 - Prob. 14ECh. 5 - Prove, that for every integer n > 1,
Ch. 5 - By integrating the binomial expansion, prove that,...Ch. 5 - Prob. 17ECh. 5 - Evaluate the sum
Ch. 5 - Sum the series by observing that
and using the...Ch. 5 - Find integers a, b, and c such that
for all m....Ch. 5 - Prob. 21ECh. 5 - Prob. 22ECh. 5 - Prob. 23ECh. 5 - Prob. 24ECh. 5 - Use a combinatorial argument to prove the...Ch. 5 - Let n and k be integers with 1 ≤ k ≤ n. Prove...Ch. 5 - Let n and k be positive integers. Give a...Ch. 5 - Let n and k be positive integers. Give a...Ch. 5 - Find and prove a formula for
where the summation...Ch. 5 - Prove that the only antichain of S = {1, 2, 3, 4}...Ch. 5 - Prove that there are only two antichains of S =...Ch. 5 - Let S be a set of n elements. Prove that, if n is...Ch. 5 - Construct a partition of the subsets of {1, 2, 3,...Ch. 5 - In a partition of the subsets of {1,2, …, n} into...Ch. 5 - A talk show host has just bought 10 new jokes....Ch. 5 - Prove the identity of Exercise 25 using the...Ch. 5 - Use the multinomial theorem to show that, for...Ch. 5 - Use the multinomial theorem to expand (x1 + x2 +...Ch. 5 - Determine the coefficient of in the expansion...Ch. 5 - What is the coefficient of in the expansion of
Ch. 5 - Prob. 41ECh. 5 - Prob. 42ECh. 5 - Prove by induction on n that, for n a positive...Ch. 5 - Prove that
where the summation extends over all...Ch. 5 - Prove that
where the summation extends over all...Ch. 5 - Use Newton’s binomial theorem to approximate .
Ch. 5 - Use Newton’s binomial theorem to approximate...Ch. 5 - Use Theorem 5.6.1 to show that, if m and n are...Ch. 5 - Use Theorem 5.6.1 to show that, if m and n are...Ch. 5 - Prob. 50ECh. 5 - Let R and S be two partial orders on the same set...
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- 5) For each function represented by an equation, make a table and plot the corresponding points to sketch the graph of the function. (a) y = 75 ()* 220 X y 200- -2 180 160 -1 140 0 120 100 1 60 80 2 3 4 x (b) y = 20 ()* 1 60 40 20 20 0 2 3 65- -1 X y 60 -2 55- 50 45 44 40 0 35- 30 1 25 2 20 20 15 3 10 5 LO 4 3-2 T -1 0 5- 4- -3- 2-arrow_forward5. Find the solution to each of the following by using an appropriate formula developed in the lecture slides: (a) + 3y = 2, y(0) = 4; (b) dy - 7y = 7, y(0) = 7; (c) 3d+6y= 5, y(0) = 0arrow_forward1. Evaluate the following improper integrals: (a) fe-rt dt; (b) fert dt; (c) fi da dxarrow_forward
- 8. Given the rate of net investment I(t) = 9t¹/2, find the level of capital formation in (i) 16 years and (ii) between the 4th and the 8th years.arrow_forward9. If the marginal revenue function of a firm in the production of output is MR = 40 - 10q² where q is the level of output, and total revenue is 120 at 3 units of output, find the total revenue function. [Hints: TR = √ MRdq]arrow_forward6. Solve the following first-order linear differential equations; if an initial condition is given, definitize the arbitrary constant: (a) 2 + 12y + 2et = 0, y(0) = /; (b) dy+y=tarrow_forward
- 4. Let A = {a, b, c, d, e, f}, B = {e, f, g, h} and C = {a, e, h,i}. Let U = {a, b, c, d, e, f, g, h, i, j, k}. • Draw a Venn Diagram to describe the relationships between these sets Find (AB) NC • Find (AC) UB Find AUBUC • Find (BC) N (A - C)arrow_forward7. A consumer lives on an island where she produces two goods x and y according to the production possibility frontier x² + y² < 200 and she consumes all the goods. Her utility function is U(x, y) = x y³. She faces an environmental constraint on her total output of both goods. The environmental constraint is given by x + y ≤20. • (a) Write down the consumer's optimization problem. (b) Write out the Kuhn-Tucker first order conditions. (c) Find the consumer's optimal consumption bundle (x*, y*).arrow_forward3. Answer the following questions: (a) Given the marginal propensity to import M'(Y) = 0.1 and the information that M = 20 when Y = 0, find the import function M(Y). (b) Given a continuous income stream at the constant rate of $1,000 per year, what will be the present value II if the income stream terminates after exactly 3 years and the discount rate is 0.04? (c) What is the present value of a perpetual cash flow of $2,460 per year, discounted at r = 8%?arrow_forward
- 5. Let A and B be arbitrary sets. Prove AnB = AUB.arrow_forward2. Answer the following questions: (a) Given the marginal-revenue function R'(Q) = 28Q - €0.3Q, find the total-revenue function R(Q). What initial condition can you introduce to definitize the constant of integration? = (b) Given the marginal propensity to consume C'(Y) 0.80.1Y-1/2 and the information that C = Y when Y = 100, find the consumption function C(Y).arrow_forwardX GG G + C td.bksblive2.com.au/bksblive2/Play... E R New Chrome available CANVAS gmetrix N notion Six big immigratio... >>> All Bookmarks 1.1 ACSF L5 SC Geometry and Measure: Vectors Vectors State the vector quantities shown on the image below. AB = CD' = A B D < C 80 esc F1 F2 F3 F4 ? Help 7arrow_forward
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