Question

What is the total number of terms in the expansion of \( (x+y)^{15} \)?

• A. \(15 \)
• B. \(16 \)
• C. \(14 \)
• D. \(30 \)

Hint:

For binomial expansion \( (a+b)^n \), always remember: Total number of terms = \(n+1\).

Answer 1

The Correct Option is B

Concept: The expansion of an expression of the form: \[ (a+b)^n \] is determined by the Binomial Theorem. According to the Binomial Theorem: \[ (a+b)^n=\sum_{r=0}^{n}\, ^nC_r a^{n-r}b^r \] An important result from this theorem is that the total number of distinct terms in the expansion is always: \[ n+1 \] where \(n\) is the exponent.

Step 1: Understanding the formula for total number of terms in binomial expansion.
For any binomial expression: \[ (a+b)^n \] the number of terms generated in complete expansion is: \[ n+1 \] This happens because the powers of the second term start from zero and continue up to \(n\). Thus there are exactly \(n+1\) separate terms.

Step 2: Applying the formula to the given expression.
The given expression is: \[ (x+y)^{15} \] Here the exponent is: \[ n=15 \] Using formula: \[ \text{Number of terms}=n+1 \] Substituting: \[ \text{Number of terms}=15+1 \] \[ =16 \]

Step 3: Writing the final conclusion.
Therefore, the expansion of \[ (x+y)^{15} \] contains exactly: \[ 16 \] distinct terms. Hence, \[ \boxed{16} \] So option (B) is correct.