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271
Kenneth Rosen Edition 7 Exercise 6.4 Question 15 (Page No. 421)
Show that $\binom{n}{k} \leq 2^{n}$ for all positive integers $n$ and all integers $k$ with $0 \leq k \leq n.$
Show that $\binom{n}{k} \leq 2^{n}$ for all positive integers $n$ and all integers $k$ with $0 \leq k \leq n.$
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273
Kenneth Rosen Edition 7 Exercise 6.4 Question 13 (Page No. 421)
What is the row of Pascal’s triangle containing the binomial coefficients $\binom{9}{k} ,\: 0 \leq k \leq 9?$
What is the row of Pascal’s triangle containing the binomial coefficients $\binom{9}{k} ,\: 0 \leq k \leq 9?$
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275
Kenneth Rosen Edition 7 Exercise 6.4 Question 11 (Page No. 421)
Give a formula for the coefficient of $x^{k}$ in the expansion of $\left(x^{2} − \frac{1}{x}\right)^{100},$ where $k$ is an integer.
Give a formula for the coefficient of $x^{k}$ in the expansion of $\left(x^{2} − \frac{1}{x}\right)^{100},$ where $k$ is an integer.
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276
Kenneth Rosen Edition 7 Exercise 6.4 Question 10 (Page No. 421)
Give a formula for the coefficient of $x^{k}$ in the expansion of $\left(x + \frac{1}{x}\right)^{100},$ where $k$ is an integer.
Give a formula for the coefficient of $x^{k}$ in the expansion of $\left(x + \frac{1}{x}\right)^{100},$ where $k$ is an integer.
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284
Kenneth Rosen Edition 7 Exercise 6.4 Question 2 (Page No. 421)
Find the expansion of $(x + y)^{5}$ using combinatorial reasoning, as in Example $1.$ using the binomial theorem.
Find the expansion of $(x + y)^{5}$using combinatorial reasoning, as in Example $1.$using the binomial theorem.
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285
Kenneth Rosen Edition 7 Exercise 6.4 Question 1 (Page No. 421)
Find the expansion of $(x + y)^{4}$ using combinatorial reasoning, as in Example $1.$ using the binomial theorem.
Find the expansion of $(x + y)^{4}$using combinatorial reasoning, as in Example $1.$ using the binomial theorem.
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288
Kenneth Rosen Edition 7 Exercise 6.3 Question 44 (Page No. 415)
How many ways are there for a horse race with four horses to finish if ties are possible? [Note: Any number of the four horses may tie.)
How many ways are there for a horse race with four horses to finish if ties are possible? [Note: Any number of the four horses may tie.)
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289
Kenneth Rosen Edition 7 Exercise 6.3 Question 43 (Page No. 415)
How many ways are there for a horse race with three horses to finish if ties are possible? [Note: Two or three horses may tie.]
How many ways are there for a horse race with three horses to finish if ties are possible? [Note: Two or three horses may tie.]
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293
Kenneth Rosen Edition 7 Exercise 6.3 Question 39 (Page No. 415)
How many license plates consisting of three letters followed by three digits contain no letter or digit twice?
How many license plates consisting of three letters followed by three digits contain no letter or digit twice?
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296
Kenneth Rosen Edition 7 Exercise 6.3 Question 36 (Page No. 414)
How many bit strings contain exactly five $0s$ and $14\:\: 1s$ if every $0$ must be immediately followed by two $1s?$
How many bit strings contain exactly five $0s$ and $14\:\: 1s$ if every $0$ must be immediately followed by two $1s?$
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297
Kenneth Rosen Edition 7 Exercise 6.3 Question 35 (Page No. 414)
How many bit strings contain exactly eight $0s$ and $10\:\: 1s$ if every $0$ must be immediately followed by a $1?$
How many bit strings contain exactly eight $0s$ and $10\:\: 1s$ if every $0$ must be immediately followed by a $1?$
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298
Kenneth Rosen Edition 7 Exercise 6.3 Question 34 (Page No. 414)
Suppose that a department contains $10$ men and $15$ women. How many ways are there to form a committee with six members if it must have more women than men?
Suppose that a department contains $10$ men and $15$ women. How many ways are there to form a committee with six members if it must have more women than men?
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299
Kenneth Rosen Edition 7 Exercise 6.3 Question 33 (Page No. 414)
Suppose that a department contains $10$ men and $15$ women. How many ways are there to form a committee with six members if it must have the same number of men and women?
Suppose that a department contains $10$ men and $15$ women. How many ways are there to form a committee with six members if it must have the same number of men and women?...
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