Chapter 15: Q2P (page 742)

In the expansion of ${(a + b)}^{n}$ (see Example 2), let $a = b = 1$ , and interpret the terms of the expansion to show that the total number of combinations of n things taken1, 2, 3, · · · , n at a time, is $2^{n} - 1$
.

Short Answer

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Answer

The total number of combinations of n things taken 1, 2, 3, · · · , n at a time, is $2^{n} - 1$ .

Step by step solution

Given Information

$a = b = 1$ in the expansion of ${(a + b)}^{n}$ .

Definition of Independent Event

When the order of arrangement is definite, the permutation is applied and when the order is not definite,combination is applied.

Find the total number of combinations of n things taken 1, 2, 3, · · ·, n at a time,

The expansion of ${(a + b)}^{n}$ is ${(a + b)}^{n} = \sum_{r = 0}^{n} C (n, r) a^{n - r} b^{r}$ .

Substitute 1 for a and 1 for b into the expansion and simplify.

${(1 + 1)}^{n} = C (n, 0) 1^{n - 0} 1^{0} + C (n, 2) 1^{n - 2} 1^{2} + 8 + C (n, n) 1^{n - n} 1^{n} 2^{n} = C (n, 0) + C (n, 1) + C (n, 2) + \otimes + C (n, n) C (n, 1) + C (n, 2) + \otimes + C (n, n) = 2^{n} - C (n, 0) C (n, 1) + C (n, 2) + \otimes + C (n, n) = 2^{n} - 1$

From the obtained simplification, it can be observed that the total number of combinations of nthings taken 1, 2, 3, · · · , nat a time, is $2^{n} - 1$ .

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In the expansion of ${(a + b)}^{n}$ (see Example 2), let $a = b = 1$ , and interpret the terms of the expansion to show that the total number of combinations of n things taken1, 2, 3, · · · , n at a time, is $2^{n} - 1$
.

Short Answer

Step by step solution

Given Information

Definition of Independent Event

Find the total number of combinations of n things taken 1, 2, 3, · · ·, n at a time,

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In the expansion of (a+b)n(see Example 2), let a=b=1, and interpret the terms of the expansion to show that the total number of combinations of n things taken1, 2, 3, · · · , n at a time, is 2n-1.

Short Answer

Step by step solution

Given Information

Definition of Independent Event

Find the total number of combinations of n things taken 1, 2, 3, · · ·, n at a time,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Engineering Physics

Electric Charge Field and Potential

Circular Motion and Gravitation

Nuclear Physics

Electricity and Magnetism

Study anywhere. Anytime. Across all devices.

In the expansion of ${(a + b)}^{n}$ (see Example 2), let $a = b = 1$ , and interpret the terms of the expansion to show that the total number of combinations of n things taken1, 2, 3, · · · , n at a time, is $2^{n} - 1$
.