Problem 14.1 | Problem 12.11

A First Course in String Theory

14.1 Counting bosonic states

12.11 Counting symmetric products

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$a^{I_1} a^{I_2} ... a^{I_n}$

$i = 1, 2, ..., n$

$I_i = 1, 2, ..., k$

$n$ is the number of $a$ ‘s.

$k$ is the number of different $a$ ‘s.

For $k=6$ and $n=9$ , one of the possible configurations is

$a^1 a^2 a^2 a^4 a^4 a^4 a^5 a^6 a^6$

The problem is equivalent to that of assigning $n$ balls to $k$ boxes.

Box 1 collects the $a$ ‘s with index 1.

Box 2 collects the $a$ ‘s with index 2.

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