Find the value of (1 + 2)^4 using the binomial theorem.

Find the value of (1 + 2)^4 using the binomial theorem.

Step 1: Identify the expression we want to calculate, which is (1 + 2)^4.
Step 2: Recognize that we can use the binomial theorem, which states that (a + b)^n = Σ (C(n, k) * a^(n-k) * b^k) for k = 0 to n.
Step 3: In our case, a = 1, b = 2, and n = 4.
Step 4: Write out the binomial expansion using the formula: (1 + 2)^4 = C(4,0) * 1^4 * 2^0 + C(4,1) * 1^3 * 2^1 + C(4,2) * 1^2 * 2^2 + C(4,3) * 1^1 * 2^3 + C(4,4) * 1^0 * 2^4.
Step 5: Calculate each term separately: C(4,0) * 1^4 * 2^0 = 1 * 1 * 1 = 1.
Step 6: Calculate the second term: C(4,1) * 1^3 * 2^1 = 4 * 1 * 2 = 8.
Step 7: Calculate the third term: C(4,2) * 1^2 * 2^2 = 6 * 1 * 4 = 24.
Step 8: Calculate the fourth term: C(4,3) * 1^1 * 2^3 = 4 * 1 * 8 = 32.
Step 9: Calculate the fifth term: C(4,4) * 1^0 * 2^4 = 1 * 1 * 16 = 16.
Step 10: Add all the terms together: 1 + 8 + 24 + 32 + 16 = 81.
Step 11: Conclude that the value of (1 + 2)^4 is 81.

Binomial Theorem – The binomial theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a non-negative integer.
Combinatorial Coefficients – The coefficients C(n, k) represent the number of ways to choose k elements from a set of n elements, which are used in the expansion.
Polynomial Expansion – The process of expanding a polynomial expression into a sum of terms, each of which is a product of a coefficient and variables raised to powers.