There are more than 1000000 ways in which $n$ identical black socks and $2 n$ identical gold socks can be arranged in a row so that there are at least 2 gold socks between any 2 black socks. The sum of the digits of the smallest possible value of $n$ is
(A) 9
(B) 10
(C) 11
(D) 12
(E) 13
Since there must be at least 2 gold socks $(G)$ between any 2 black socks $(B)$, then we start by placing the $n$ black socks with exactly 2 gold socks in between each pair:

## $B G G B G G B G G B \cdots B G G B$

Since there are $n$ black socks, then there are $n-1$ "gaps" between them and so $2(n-1)$ or $2 n-2$ gold socks have been used.

This means that there are 2 gold socks left to place. There are $n+1$ locations in which these socks can be placed: either before the first black sock, after the last black sock, or in one of the $n-1$ gaps.

These 2 socks can either be placed together in one location or separately in two locations.

If the 2 socks are placed together, they can be placed in any one of the $n+1$ locations, and so there are $n+1$ ways to do this. (The placement of the gold socks within these locations does not matter as the gold socks are all identical.)

The other possibility is that the 2 gold socks are placed separately in two of these $n+1$ spots. There are $n+1$ possible locations for the first of these socks. For each of these, there are then $n$ possible locations for the second of these socks (any other than the location of the first sock). Since these two socks are identical, we have double-counted the total number of possibilities, and so there are $\frac{1}{2}(n+1) n$ ways for these two gold socks to be placed.

In total, there are $(n+1)+\frac{1}{2}(n+1) n=(n+1)\left(1+\frac{1}{2} n\right)=\frac{1}{2}(n+1)(n+2)$ ways in which the socks can be arranged.

We want to determine the smallest positive integer $n$ for which this total is greater than 1000000 .

This is equivalent to determining the smallest positive integer $n$ for which $(n+1)(n+2)$ is greater than 2000000.

We note that $(n+1)(n+2)$ increases as $n$ increases, since each of $n+1$ and $n+2$ is positive and increasing, and so their product is increasing.

When $n=1412$, we have $(n+1)(n+2)=1997982$.

When $n=1413$, we have $(n+1)(n+2)=2000810$.

Since $(n+1)(n+2)$ is increasing, then $n=1413$ is the smallest positive integer for which $(n+1)(n+2)$ is greater than 2000000 , and so it is the smallest positive integer for which there are more than 1000000 arrangements of the socks.

The sum of the digits of $n=1413$ is $1+4+1+3=9$.

ANSWER: (A)