Question

Let $n = \sum_{r=0}^{10} (-1)^r \binom{10}{r} \left(\frac{2}{3}\right)^{2r} 3^{20}$. Which one of the following statements is TRUE?

• A. $n$ is divisible by 5
• B. $n$ is divisible by 6
• C. $n$ is divisible by 8
• D. $n$ is divisible by 9

Hint:

When dealing with summations involving $\binom{N}{r}$, look to factor out terms independent of $r$ first.
Matching the remaining terms to the binomial identity $(1+x)^N$ makes simplifying such expressions very straightforward.

Answer 1

The Correct Option is A

Step 1 : Understanding the Question:
The question asks to simplify a summation expression containing binomial coefficients and determine which of the given options correctly describes its divisibility.

Step 2 : Key Formulas and Approach:
The Binomial Theorem states that:
\[ (1 + x)^N = \sum_{r=0}^{N} \binom{N}{r} x^r \]
We will rewrite the terms inside the summation to match the binomial expansion formula, simplify the expression for $n$, and test its divisibility.

Step 3 : Detailed Explanation:
Let us write down the given expression for $n$:
\[ n = \sum_{r=0}^{10} (-1)^r \binom{10}{r} \left(\frac{2}{3}\right)^{2r} 3^{20} \]
We can simplify the term $\left(\frac{2}{3}\right)^{2r}$:
\[ \left(\frac{2}{3}\right)^{2r} = \left[\left(\frac{2}{3}\right)^2\right]^r = \left(\frac{4}{9}\right)^r \]
Now, substitute this back into the expression:
\[ n = \sum_{r=0}^{10} \binom{10}{r} (-1)^r \left(\frac{4}{9}\right)^r 3^{20} \]
Since $3^{20}$ is independent of the summation index $r$, we can factor it outside the summation:
\[ n = 3^{20} \sum_{r=0}^{10} \binom{10}{r} \left(-\frac{4}{9}\right)^r \]
The summation term is exactly the binomial expansion of $(1 + x)^{10}$ with $x = -\frac{4}{9}$:
\[ \sum_{r=0}^{10} \binom{10}{r} \left(-\frac{4}{9}\right)^r = \left(1 - \frac{4}{9}\right)^{10} = \left(\frac{5}{9}\right)^{10} \]
Substitute this back into the equation for $n$:
\[ n = 3^{20} \cdot \left(\frac{5}{9}\right)^{10} \]
Expressing the denominator as a power of 3 ($9^{10} = (3^2)^{10} = 3^{20}$):
\[ n = 3^{20} \cdot \frac{5^{10}}{3^{20}} \]
The terms $3^{20}$ cancel out, leaving:
\[ n = 5^{10} \]
Since $n = 5^{10}$, it is clearly a power of 5, which means it is divisible by 5, but not by 6, 8, or 9.

Step 4 : Final Answer:
The simplified value of $n$ is $5^{10}$, which is divisible by 5.
This corresponds to Option (A).