To find the sixth term in the binomial expansion of the expression (5y + 3)^10, we can use the Binomial Theorem. The theorem states that for any binomial raised to the power of n, the expansion can be expressed as:
(a + b)^n = Σ [C(n, k) * a^(n-k) * b^k] where k = 0 to n
Here, in our expression (5y + 3)^10:
- a = 5y
- b = 3
- n = 10
We need to find the sixth term, which corresponds to k = 5 (since we start counting from k = 0). Using the formula for the term:
Term(k + 1) = C(n, k) * a^(n-k) * b^k
For the sixth term:
- C(10, 5) is the binomial coefficient for choosing 5 items from 10 items, calculated as:
- C(10, 5) = 10! / (5! * (10 – 5)!) = 252
- a^(10-5) = (5y)^5 = 5^5 * y^5 = 3125y^5
- b^5 = 3^5 = 243
Putting it all together:
Sixth Term = C(10, 5) * (5y)^5 * (3)^5
Sixth Term = 252 * 3125y^5 * 243
Now, simplifying this gives us:
Sixth Term = 252 * 3125 * 243 * y^5
Thus, the expression that represents the sixth term in the binomial expansion of (5y + 3)^10 is:
252 * 3125 * 243 * y^5