Chapter 12: Q. 12.68. (page 1208)

Evaluate each binomial coefficient,
$(a) (\begin{matrix} 2 \\ 1 \end{matrix}) (b) (\begin{matrix} 11 \\ 11 \end{matrix}) (c) (\begin{matrix} 9 \\ 6 \end{matrix}) (d) (\begin{matrix} 6 \\ 5 \end{matrix})$

Short Answer

Expert verified

Evaluated form of each binomial coefficient,

$(a) 2 (b) 1 (c) 84 (d) 6$

Step by step solution

Part (a) Step 1. Evaluate the binomial by using the formula.

Use the definition of a binomial coefficient,

$(\begin{matrix} n \\ r \end{matrix}) = \frac{n!}{r! (n - r)!}$

Use this formula to solve the given coefficient.

$(\begin{matrix} 2 \\ 1 \end{matrix}) = \frac{2!}{1! (2 - 1)!} = \frac{2!}{1! (1)!} = \frac{2 \times 1}{1 \times 1} = 2$

Part (b) Step 1. Evaluate the binomial by using the formula.

Use the formula to solve the given coefficient.

$(\begin{matrix} 11 \\ 11 \end{matrix}) = \frac{11!}{11! (11 - 11)!} = \frac{11!}{11! (0)!} = \frac{11!}{11! (1)} = \frac{11!}{11!} = 1$

Part (c) Step 1. Evaluate the binomial by using the formula.

Use the formula to solve the given coefficient.

$(\begin{matrix} 9 \\ 6 \end{matrix}) = \frac{9!}{6! (9 - 6)!} = \frac{9!}{6! (3)!} = \frac{9 \times 8 \times 7 \times 6!}{6! \times 3 \times 2 \times 1} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$

Part (d) Step 1. Evaluate the binomial by using the formula.

Use the formula to solve the given coefficient.

$(\begin{matrix} 6 \\ 5 \end{matrix}) = \frac{6!}{5! (6 - 5)!} = \frac{6!}{5! (1)!} = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1} = 6$

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Evaluate each binomial coefficient,
$(a) (\begin{matrix} 2 \\ 1 \end{matrix}) (b) (\begin{matrix} 11 \\ 11 \end{matrix}) (c) (\begin{matrix} 9 \\ 6 \end{matrix}) (d) (\begin{matrix} 6 \\ 5 \end{matrix})$

Short Answer

Step by step solution

Part (a) Step 1. Evaluate the binomial by using the formula.

Part (b) Step 1. Evaluate the binomial by using the formula.

Part (c) Step 1. Evaluate the binomial by using the formula.

Part (d) Step 1. Evaluate the binomial by using the formula.

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Evaluate each binomial coefficient,a21b1111c96d65

Short Answer

Step by step solution

Part (a) Step 1. Evaluate the binomial by using the formula.

Part (b) Step 1. Evaluate the binomial by using the formula.

Part (c) Step 1. Evaluate the binomial by using the formula.

Part (d) Step 1. Evaluate the binomial by using the formula.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Statistics

Theoretical and Mathematical Physics

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Evaluate each binomial coefficient,
$(a) (\begin{matrix} 2 \\ 1 \end{matrix}) (b) (\begin{matrix} 11 \\ 11 \end{matrix}) (c) (\begin{matrix} 9 \\ 6 \end{matrix}) (d) (\begin{matrix} 6 \\ 5 \end{matrix})$