How many ways can 26 letters be divided into 13 pairs? The question is: a plug board has 26 letters, and there are 13 cables. The cables connect all possible pair of letters.
How many possible configurations does the plug board provide? In other words, how many ways can 26 letters be divided into 13 pairs?
The result I found by calculation and some logic is:
$$25\cdot 23\cdot 21\cdot 19\cdot 17\cdot 15\cdot 13\cdot 11\cdot 9\cdot 7\cdot 5\cdot 3 = 7 905 853 580 625.$$
Is this correct?
How is the mathematical algorithm and formula to calculate this and similar problems?

Another approach is: the first pair can be selected in ${26\choose 2}$ ways, but we are not interested in the position of the pair among the $13$ pairs, hence we divide it by ${13\choose 1}$:
$$\frac{{26\choose 2}}{{13\choose 1}}$$
Similarly, for other pairs. 
At the end we get:
$$\frac{{26\choose 2}}{{13\choose 1}}\cdot \frac{{24\choose 2}}{{12\choose 1}}\cdot\frac{{22\choose 2}}{{11\choose 1}}\cdot\frac{{20\choose 2}}{{10\choose 1}}\cdot \frac{{18\choose 2}}{{9\choose 1}}\cdot\frac{{16\choose 2}}{{8\choose 1}}\cdot\frac{{14\choose 2}}{{7\choose 1}}\cdot\frac{{12\choose 2}}{{6\choose 1}}\cdot\frac{{10\choose 2}}{{5\choose 1}}\cdot\frac{{8\choose 2}}{{4\choose 1}}\cdot\frac{{6\choose 2}}{{3\choose 1}}\cdot\frac{{4\choose 2}}{{2\choose 1}}\cdot\frac{{2\choose 2}}{{1\choose 1}},$$
which is what you have.
