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.

From Prime Factors of $52!$:


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

From Prime Factors of $39!$:


 * $39! = 2^{35} \times 3^{18} \times 5^8 \times 7^5 \times 11^3 \times 13^3 \times 17^2 \times 19^2 \times 23 \times 29 \times 31 \times 37$

From Prime Factors of $13!$:


 * $13! = 2^{10} \times 3^5 \times 5^2 \times 7 \times 11 \times 13$

For each prime factor $p$ of $\dbinom {52} {13}$, let $a_p$ be the integer such that:
 * $p^{a_p} \mathrel \backslash \dbinom {52} {13}$
 * $p^{a_p + 1} \nmid \dbinom {52} {13}$

Taking the prime factors in turn:

Finally:
 * $a_{41} = a_{43} = a_{47} = 1$

Hence the result.