What is the expression for the sixth term in the binomial expansion of (5y + 3)^{10}?

To find the sixth term in the binomial expansion of the expression (5y + 3)¹⁰, we can use the binomial theorem. The binomial theorem states that:

(a + b)ⁿ =

= (nCr) * a^n-r * b^r, where nCr is the binomial coefficient, n is the power, a is the first term, b is the second term, and r is the term number (starting from 0).

In our case, we have:

• a = 5y
• b = 3
• n = 10

We are looking for the sixth term, which corresponds to r = 5 (since we start counting from 0). Now we can use the formula to find the sixth term:

Term(5) = 10C5 * (5y)^(10-5) * (3)⁵

Now we need to compute each part:

1. Calculate the binomial coefficient: 10C5 = 252
2. Calculate (5y)⁵ = 5⁵ * y⁵ = 3125y⁵
3. Calculate (3)⁵ = 243

Putting it all together:

Term(5) = 252 * 3125y⁵ * 243

Now let’s multiply these values:

252 * 243 = 61236

Finally, we can find the sixth term:

Term(5) = 61236 * 3125y⁵

So, the sixth term in the binomial expansion of (5y + 3)¹⁰ is:

191250000y⁵