Question

(x^1+x^2+............+x^100)^500 if you apply binomial theorem how many digits there will be

Answer 1

Answer on Question #81709 – Math– Combinatorics | Number Theory Question

(x^1+x^2+...+x^100)^500 if you apply binomial theorem how many digits there will be

Solution

If by digit is meant any summand and coefficients are not put together (i.e. form $x^{500} + x^{500} + \ldots + x^{500} + x^{501} + \ldots + x^{501} + \ldots + x^{50000} + \ldots + x^{50000}$) then the solution is following.

There are 100 ways to choose the digit from the first multiplier $x + x^2 + \ldots + x^{100}$, 100 ways to choose the digit from the second multiplier, and so on up to 100 ways to choose the digit from the $500^{\text{th}}$ multiplier. Then the total number of ways is

$$\frac{100 \cdot 100 \cdots 100}{500 \text{ times}} = 100^{500}, \text{ it is the number of digits}.$$

I thought that the question is about the number of digits which arise as powers (because the binomial theorem gives a form of sum with coefficients).

Then it was my solution:

The minimum power of $x$ in the expansion of $(x + x^2 + x^3 + \ldots + x^{100})^{500}$ is 500: $x \cdot x \cdot x \cdot \ldots \cdot x = x^{500}$, the maximum power of $x$ in the expansion is 50000: $x^{100} \cdot x^{100} \cdot \ldots \cdot x^{100} = (x^{100})^{500} = x^{50000}$.

Let us prove that for any $n \in \mathbb{Z}$: $500 \leq n < 50000x^n$ is present in the expansion. We have

$$(x + x^2 + x^3 + \ldots + x^{100})^{500} = x^{500}(1 + x + x^2 + \ldots + x^{99})^{500}$$

Then we have to prove that for any $0 \leq m < 49500x^m$ is present in the expansion of $(1 + x + x^2 + \ldots + x^{99})^{500}$.

Divide $m$ by 99:

$$m = 99k + l$$

where

$$0 \leq k < 500, 0 \leq l < 99.$$

Then $x^m$ can be represented as $x^m = \underbrace{x^{99} \cdot x^{99} \cdot \ldots \cdot x^{99}}_{k \text{ times}} \cdot x^l \cdot \underbrace{1 \cdot 1 \cdot \ldots \cdot 1}_{500 - 1 - k \geq 0 \text{ times}}$, thus it is present in the expansion.

Then the number of digits is 50000-500+1=49501 (we add 1 since we need to include both numbers 500 and 50000, and 50000-500 is number of numbers between 500 and 50000 not including 50000).

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