Binomial Coefficient/Examples/Number of Bridge Hands/Prime Factors

Example of Factorial

The prime decomposition of the number of bridge hands is given as:

$\dbinom {52} {13} = 2^4 \times 5^2 \times 7^2 \times 17 \times 23 \times 41 \times 43 \times 47$

Proof

We have by definition of binomial coefficient:

$\dbinom {52} {13} = \dfrac {52!} {13! \, 39!}$

Thus it is necessary to determine the prime factors of each of the contributing factorials here.

$52! = 2^{49} \times 3^{23} \times 5^{12} \times 7^8 \times 11^4 \times 13^4 \times 17^3 \times 19^2 \times 23^2 \times 29 \times 31 \times 37 \times 41 \times 43 \times 47$
$39! = 2^{35} \times 3^{18} \times 5^8 \times 7^5 \times 11^3 \times 13^3 \times 17^2 \times 19^2 \times 23 \times 29 \times 31 \times 37$
$13! = 2^{10} \times 3^5 \times 5^2 \times 7 \times 11 \times 13$

For each prime factor $p$ of $\dbinom {52} {13}$, let $a_p$ be the integer such that:

$p^{a_p} \divides \dbinom {52} {13}$
$p^{a_p + 1} \nmid \dbinom {52} {13}$

Taking the prime factors in turn:

 $\ds a_2$ $=$ $\ds 49 - 35 - 10$ $\ds$ $=$ $\ds 4$
 $\ds a_3$ $=$ $\ds 23 - 18 - 5$ $\ds$ $=$ $\ds 0$
 $\ds a_5$ $=$ $\ds 12 - 8 - 2$ $\ds$ $=$ $\ds 2$
 $\ds a_7$ $=$ $\ds 8 - 5 - 1$ $\ds$ $=$ $\ds 2$
 $\ds a_{11}$ $=$ $\ds 4 - 3 - 1$ $\ds$ $=$ $\ds 0$
 $\ds a_{13}$ $=$ $\ds 4 - 3 - 1$ $\ds$ $=$ $\ds 0$
 $\ds a_{17}$ $=$ $\ds 3 - 2$ $\ds$ $=$ $\ds 1$
 $\ds a_{19}$ $=$ $\ds 2 - 2$ $\ds$ $=$ $\ds 0$
 $\ds a_{23}$ $=$ $\ds 2 - 1$ $\ds$ $=$ $\ds 1$
 $\ds a_{29}$ $=$ $\ds 1 - 1$ $\ds$ $=$ $\ds 0$
 $\ds a_{31}$ $=$ $\ds 1 - 1$ $\ds$ $=$ $\ds 0$
 $\ds a_{37}$ $=$ $\ds 1 - 1$ $\ds$ $=$ $\ds 0$

Finally:

$a_{41} = a_{43} = a_{47} = 1$

Hence the result.

$\blacksquare$