Solving for the Deck Size: When 5-Card Hands Equal 2-Card Hands

In this article, we delve into a fascinating problem from combinatorics: finding the size of a deck such that the number of 5-card hands matches the number of 2-card hands. This exploration will illuminate the principles of combinations and provide a step-by-step solution.

Principle of Combinations

The number of ways to choose k cards from n cards is given by the combination formula:

Combination Formula

( binom{n}{k} frac{n!}{k!(n-k)!} )

Where:

( binom{n}{k} ) represents the number of ways to choose k cards from n cards. ( n! ) denotes the factorial of n. ( k! ) denotes the factorial of k. ( (n-k)! ) denotes the factorial of n-k.

We need to solve the equation:

Equation for Deck Size

( binom{n}{5} binom{n}{2} )

Deriving the Equation

Substitute the combination formula into the equality:

Step 1: Substituting the Combination Formula

( frac{n!}{5!(n-5)!} frac{n!}{2!(n-2)!} )

Cancel ( n! ) from both sides, assuming ( n geq 5 ):

Step 2: Canceling ( n! )

( frac{1}{5!(n-5)!} frac{1}{2!(n-2)!} )

Cross-multiplying gives:

Step 3: Cross-Multiplying

( 2!(n-2)! 5!(n-5)! )

Substitute the factorials:

Step 4: Substituting the Factorials

( 2(n-2)(n-3) 120(n-5)(n-4)(n-3)(n-2) )

Since ( (n-2)(n-3) ) is common on both sides, we can simplify:

Step 5: Simplifying

( 2 120(n-5)(n-4) )

Divide both sides by 120:

( frac{2}{120} (n-5)(n-4) )

This simplifies to:

( frac{1}{60} (n-5)(n-4) )

Now, we need to find integer solutions for ( n ) starting from 5:

Testing Integer Solutions for ( n )

For ( n 5 ):
( 5-25-3) 0, which is not equal to 60.

For ( n 6 ):
( 6-26-3) 24, which is not equal to 60.

For ( n 7 ):
( 7-27-3) 60, which is equal to 60.

Hence, the value of ( n ) that satisfies the condition is boxed{7}.

For a ( 7-)card deck, the number of 5-card hands equals the number of 2-card hands, as ( binom{7}{5} binom{7}{2} 21 ). Each unique way to have five cards corresponds to a unique way to have all but two cards, and vice versa.

The same principle applies to a standard poker deck. The number of 47-card hands is the same as the number of 5-card hands, given that ( binom{52}{5} binom{52}{47} 2,598,960 ).

This exploration demonstrates the beauty of combinatorial mathematics and provides a practical application to the problem of deck size in card games.