Final answer:
The coefficient of x^6y^7 in the expansion of (x+3y)^13 is calculated using the binomial theorem, resulting in 1716 * 3^7.
Step-by-step explanation:
To find the coefficient of x^6y^7 in the expression (x+3y)^13, we can use the binomial theorem, which allows us to expand a binomial raised to a power. The term we are looking for will come from the part of the expansion where the powers of x and y add up to 13 (since 6+7=13). The binomial coefficient for this term can be determined using the combination formula C(n, k) = n! / (k! * (n-k)!), where n is the power to which the binomial is raised and k is the term's index (starting from 0).
In this case, to find the coefficient, we take the 6th term of the expansion (since we start counting from 0, the 6th term actually has x raised to the 6th power). The combination formula gives us C(13, 6) for the binomial coefficient. So, we have:
C(13, 6) * x^6 * (3y)^7
Now, C(13, 6) equals 1716, and (3y)^7 is 3^7 * y^7. Hence, our coefficient is:
1716 * 3^7
Therefore, the coefficient of x^6y^7 when we have (x+3y)^13 is 1716 * 3^7.