Find the coefficient of 𝑥3 in (1 + 14 𝑥)6

To find the coefficient of x^3 in (1 + 14x)^6, we can use the binomial theorem, which states that:

(a + b)^n = Σ (from k=0 to n) [C(n, k) * a^(n-k) * b^k]

where C(n, k) is the binomial coefficient "n choose k".

In this case, a = 1, b = 14x, and n = 6. We want to find the term where the power of x is 3, which means k = 3.

So, we need to calculate C(6, 3) * 1^(6-3) * (14x)^3.

C(6, 3) = 6! / [3!(6-3)!] = 20

1^(6-3) = 1

(14x)^3 = 2744x^3

So, the coefficient of x^3 in (1 + 14x)^6 is 20 * 1 * 2744 = 54880.