The number of dissimilar terms in the expansion of(a+b)nis n + 1, ther...
To find the number of dissimilar terms in the expansion of (a + b)^n, we can use the concept of binomial coefficients. The binomial coefficient, denoted by C(n, r), represents the number of ways to choose r items from a set of n items without regard to their order.
In the expansion of (a + b)^n, the power of a starts from n and decreases by 1 with each term, while the power of b starts from 0 and increases by 1 with each term. The sum of the powers of a and b in each term is always n.
Now, let's consider the expansion of (a + b + c)^12.
**Step 1: Determine the number of terms in the expansion**
The number of terms in the expansion of (a + b + c)^12 can be found using the formula (n + 1), where n is the power of the binomial.
Number of terms = (12 + 1) = 13
**Step 2: Determine the powers of a, b, and c in each term**
In the expansion of (a + b + c)^12, the powers of a, b, and c in each term will be non-negative integers that satisfy the condition a + b + c = 12.
One way to determine the powers is to consider all possible combinations of powers that add up to 12. We can use the concept of binomial coefficients to find these combinations.
For example, the term a^2b^3c^7 can be represented as C(12, 2) * C(10, 3) * C(7, 7), where C(12, 2) represents the number of ways to choose 2 items from a set of 12 items, C(10, 3) represents the number of ways to choose 3 items from a set of 10 items, and C(7, 7) represents the number of ways to choose 7 items from a set of 7 items.
**Step 3: Calculate the number of dissimilar terms**
To determine the number of dissimilar terms, we need to count the number of unique combinations of powers.
In the expansion of (a + b + c)^12, there are 13 terms. Each term corresponds to a unique combination of powers of a, b, and c. Therefore, the number of dissimilar terms is 13.
Hence, the correct answer is option A) 91.