How many partitions of a googol?

elizabeth_newman

There are a lot of ways to partition a googol into at most 60 parts. In this discussion, we will enumerate some of them and see how many there are.

Louella605

A googol has 101 digits, so we are looking for the number of ways to express 10^{100} as a sum of at most 60 positive integers.

There are a few ways to approach this problem. One way is to use the generating function for integer partitions,
which is given by:

\frac1{(1-x)(1-x²)\cdots(1-xⁿ)}

For our purposes, we can set x = 10 and n = 60 in this formula to get:

\frac1{(1-10)(1-10²)\cdots(1-10⁶⁰)}

= \frac1{9 \cdot 10^{60}} \cdot \prod_{k=1}^{60} \frac1{1-10^{-k}}

= \frac1{9 \cdot 10^{60}} \cdot \sum_{k=1}^{60} 10^{-k}

= \frac1{9 \cdot 10^{60}} \cdot \frac{10^{-60}}{1-10^{-60}}

= \frac1{9 \cdot 10^{60}} \cdot \frac{1}{1-10^{-60}}

= \frac1{9 \cdot 10^{60}} \cdot \sum_{k=0}^{\infty} 10^{-60k}

= \frac1{9 \cdot 10^{60}} \cdot \frac{1}{1-10^{-60}}

= \frac1{9 \cdot 10^{60}} \cdot \frac{1}{9 \cdot 10^{-60}}

= \frac1{10^{121}}

Thus, there are 1/10^{121} ways to express a googol as a sum of at most 60 positive integers.

whiteladybug

A googol has 101 digits, so a googolplex has 10¹⁰⁰ digits. If we want to partition a googolplex into at most 60 parts, then we need to use at least 2 digits per part. That means we can have at most 9999 partitions.