Sure! Let's delve into the problem step-by-step using Pascal's Triangle to find the coefficient of the third term in the expansion of [tex]\((a + b)^6\)[/tex].
1. Understand Pascal’s Triangle: Pascal’s Triangle is a triangular array of numbers where the entries in each row are the binomial coefficients from the expansion of [tex]\((a + b)^n\)[/tex].
2. Generate Pascal’s Triangle up to row 6:
- Row 0: [tex]\(1\)[/tex]
- Row 1: [tex]\(1, 1\)[/tex]
- Row 2: [tex]\(1, 2, 1\)[/tex]
- Row 3: [tex]\(1, 3, 3, 1\)[/tex]
- Row 4: [tex]\(1, 4, 6, 4, 1\)[/tex]
- Row 5: [tex]\(1, 5, 10, 10, 5, 1\)[/tex]
- Row 6: [tex]\(1, 6, 15, 20, 15, 6, 1\)[/tex]
3. Locate the Coefficient:
- In Pascal’s Triangle, the coefficients for [tex]\((a+b)^n\)[/tex] appear in row [tex]\(n\)[/tex].
- For [tex]\((a + b)^6\)[/tex], we look at row 6: [tex]\(1, 6, 15, 20, 15, 6, 1\)[/tex].
4. Identify the Term Position:
- The terms in the expansion of [tex]\((a + b)^6\)[/tex] are ordered starting with [tex]\( (a^6 b^0) \)[/tex]. Each subsequent term is formed by decreasing the power of [tex]\(a\)[/tex] and increasing the power of [tex]\(b\)[/tex].
- The general form for the k-th term (where k starts from 0) is [tex]\( \binom{6}{k} a^{6-k} b^k \)[/tex].
- The third term corresponds to [tex]\( k = 2 \)[/tex] (since counting starts from 0).
5. Extract the Coefficient:
- The coefficient for the third term (when [tex]\(k = 2\)[/tex]) in [tex]\((a + b)^6\)[/tex] is the entry in the 3rd position of row 6.
- Row 6 (1, 6, 15, 20, 15, 6, 1), hence the third term’s coefficient is 15.
So, the coefficient of the third term in the expansion of [tex]\((a + b)^6\)[/tex] is [tex]\(15\)[/tex].
[tex]\(\boxed{15}\)[/tex]
