### Instruction: What is the value of $101^{4} - 4 \cdot 101^{3} + 6 \cdot 101^2 - 4 \cdot 101 + 1$?  (No calculators!).### Response: This looks like a very large expression, but I notice that it has a familiar pattern.
It is of the form $a^4 - 4a^3 + 6a^2 - 4a + 1$, where $a = 101$.
I recall that this is actually a special case of a binomial expansion.
It is the same as $(a - 1)^4$, since $(a - 1)^4 = a^4 - 4a^3 + 6a^2 - 4a + 1$ by the binomial theorem or by multiplying out the brackets.
So, I can rewrite the expression as $(101 - 1)^4 = 100^4$.
Now, this is much easier to calculate, since 100 is a nice round number.
I know that $100^2 = 10,000$, so $100^4 = (100^2)^2 = 10,000^2 = 100,000,000$.