Question

If the number of terms in the binomial expansion of \((2x + 3)^n\) is 22, then the value of \(n\) is:

• A. 6
• B. 7
• C. 9
• D. 8

Hint:

In a binomial expansion, the number of terms is equal to \(n + 1\), where \(n\) is the exponent in the expansion.

Answer 1

The Correct Option is B

If the number of terms in the binomial expansion of \( (2x + 3)^n \) is 22, then the value of \( n \) is:

Step 1: Formula for the number of terms in a binomial expansion
In a binomial expansion of the form \( (a + b)^n \), the number of terms in the expansion is \( n + 1 \). This is because the expansion involves terms from \( k = 0 \) to \( k = n \). Therefore, the number of terms in the expansion of \( (2x + 3)^n \) is: \[ \text{Number of terms} = n + 1 \]

Step 2: Use the given information
We are told that the number of terms is 22. Therefore, we have the equation: \[ n + 1 = 22 \] Solving for \( n \): \[ n = 22 - 1 = 21 \]

Step 3: Conclusion
Therefore, the value of \( n \) is \( \boxed{21} \).