We are tasked with finding the number of different dissimilar terms in the following expression:

$(1 + x)^{2012} + (1 + x^2)^{2011} + (1 + x^3)^{2010}$

Breaking it down term by term:

1. $(1 + x)^{2012}$:

   • This is a binomial expansion of the form $(a + b)^n$, which gives terms of the form: $\binom{2012}{k} x^k \quad \text{for} \quad k = 0, 1, 2, \dots, 2012$
   • Therefore, the exponents of $x$ in this expansion will range from $0$ to $2012$.
   • So, the possible powers of $x$ in this part are: $x^0, x^1, x^2, \dots, x^{2012}$.

2. $(1 + x^2)^{2011}$:

   • Similarly, this is a binomial expansion of the form $(a + b)^n$, but here, the second term is $x^2$.
   • The terms are of the form: $\binom{2011}{k} (x^2)^k = \binom{2011}{k} x^{2k} \quad \text{for} \quad k = 0, 1, 2, \dots, 2011$
   • The powers of $x$ will now range from $x^0, x^2, x^4, \dots, x^{4022}$ (because the highest power is $2 \times 2011 = 4022$).

3. $(1 + x^3)^{2010}$:

   • This binomial expansion involves $x^3$, and the terms will be of the form: $\binom{2010}{k} (x^3)^k = \binom{2010}{k} x^{3k} \quad \text{for} \quad k = 0, 1, 2, \dots, 2010$
   • The powers of $x$ here will be $x^0, x^3, x^6, \dots, x^{6030}$ (since the highest power is $3 \times 2010 = 6030$).

Total distinct terms:

Now, we combine the possible powers of $x$ from all three expansions. We have:

• Powers from $(1 + x)^{2012}$: $x^0, x^1, x^2, \dots, x^{2012}$
• Powers from $(1 + x^2)^{2011}$: $x^0, x^2, x^4, \dots, x^{4022}$
• Powers from $(1 + x^3)^{2010}$: $x^0, x^3, x^6, \dots, x^{6030}$

We need to find the total number of distinct exponents, so we consider the union of these three sets of exponents:

• The first set includes all integers from 0 to 2012.
• The second set includes all even integers from 0 to 4022.
• The third set includes all multiples of 3 from 0 to 6030.

To count the distinct powers of $x$, we need to consider how these overlap and combine.

Step-by-step counting:

• Integers from 0 to 2012: This contributes 2013 distinct terms $\{0, 1, 2, \dots, 2012\}$.
• Even integers from 0 to 4022: These overlap with the previous set. There are $1007$ even integers up to 2012, which have already been counted. The remaining even integers from 2014 to 4022 contribute $1005$ new distinct terms.
• Multiples of 3 from 0 to 6030: We already have multiples of 3 within the first two sets (such as 0, 3, 6, etc.). There are 671 multiples of 3 up to 2012, and 335 multiples of 3 from 2013 to 4022. The remaining multiples of 3 from 4023 to 6030 contribute additional terms.

Combining all the contributions and removing overlaps, the total number of distinct terms is:

$2013 + 1005 + 670 = 3688$

Thus, the number of different dissimilar terms in the given sum is 3688.

Would you like further details or have any other questions?

Here are 5 related questions you may find interesting:

1. How would the result change if the powers in the second binomial were $x^4$ instead of $x^2$?
2. Can you find the number of distinct terms in $(1 + x^2)^{2011}$ alone?
3. What would the number of terms be if all powers were increased by 1 in each binomial?
4. How would the distribution of powers change if $x^n$ was used in the expansions?
5. What is the effect of subtracting $(1 + x^2)^{2011}$ from the original sum?

Tip: When combining terms with different powers, always look out for overlaps to avoid over-counting terms.