Final answer:
The coefficient of a⁴b⁴ in the expansion of (a+b)⁸ is 70.
Step-by-step explanation:
The coefficient of a⁴b⁴ in the expansion of (a+b)⁸ can be found using the binomial theorem. The binomial theorem states that (a + b)ⁿ = C(n,0)aⁿb⁰ + C(n,1)aⁿ⁻¹b¹ + C(n,2)aⁿ⁻²b² + ... + C(n,n-1)a¹bⁿ⁻¹ + C(n,n)a⁰bⁿ, where C(n,r) is the binomial coefficient.
In this case, n = 8 and we are looking for the coefficient of a⁴b⁴. So, the coefficient can be found by substituting n = 8, r = 4, and evaluating C(8,4).
C(8,4) = (8!)/(4!(8-4)!) = (8!)/(4!4!) = (8 x 7 x 6 x 5)/(4 x 3 x 2 x 1) = 70.
Therefore, the coefficient of a⁴b⁴ in the expansion of (a+b)⁸ is 70.