The Binomial Theorem says that for any positive integer n and any real numbers x and y: Σ-o (R)aty"-k = (z + y)" kn-k (x + y)" k=0 Use the binomial theorem to determine the value of ΣR=o (R) (-3)* k=0
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
discrete math
![## Understanding the Binomial Theorem
The Binomial Theorem states that for any positive integer \( n \) and any real numbers \( x \) and \( y \):
\[
\sum_{k=0}^n \binom{n}{k} x^k y^{n-k} = (x + y)^n
\]
Where \(\binom{n}{k}\) denotes a binomial coefficient (also known as "n choose k").
### Application of the Binomial Theorem
Use the Binomial Theorem to determine the value of the following summation:
\[
\sum_{k=0}^n \binom{n}{k} (-3)^k
\]
### Multiple Choice Options
- \(3^n\)
- \(2^n\)
- None of the given choices.
- \((-3)^n\)
- \(4^n\)
- \((-2)^n\)
- \((-4)^n\)
### Explanation:
To determine which expression matches the given summation, apply the binomial theorem again. In the given summation, set \(x = -3\) and \(y = 1\):
\[
\sum_{k=0}^n \binom{n}{k} (-3)^k = (1 + (-3))^n = (-2)^n
\]
Therefore, the correct answer is:
\[
\boxed{(-2)^n}
\]
### Answer:
\(\mathbf{(-2)^n}\)
This content is designed to help students understand how to apply the binomial theorem to solve problems involving binomial expansions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5f558a7-14fc-4024-84d6-4debb1adc6f6%2Fc54dbea1-1b1d-4074-9fa1-f767807cfcfd%2Fu55nvfl_processed.png&w=3840&q=75)
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