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How many different 7 card hands can be chosen from a standard 52 card deck?

This question aims to find how seven standard cards can be chosen from a deck of fifty-two cards. Combination can be used to find the number of ways in which 7 hand cards can be chosen from a set of 52 card decks as the order is not specified.

Combination is the number of possible ways of arranging the selected objects from the total objects without repeating. It is expressed by a capital C.

\[ n C _ r  = \frac { n ! } { ( n – r ) ! r ! } \]

Where n is the total number of objects and r is the number of selected objects and ” ! ” is the symbol of factorial.

Expert Answer

According to the combination formula:

\[ 52 C _ 7 = C (  n  ,  r  )  =  C (  52  ,  7  ) \]

\[ 52 C  _ 7 =  \frac  {  52 !  }  {  7 !  (  52  –  7 ) ! } \]

\[  52  C _ 7 = \frac { 52 ! } { 7 ! \times   45 !  }  \]

\[  52  C _ 7 = \frac  { 52 \times 51 \times 50 \times 49 \times 48 \times 47 \times 46 \times 45 ! } { 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \times   45 !  }  \]

By simplifying the above equation:

\[  52  C _ 7 = \frac  { ( 26 \times 2 ) \times ( 17 \times 3 ) \times ( 10 \times 5 ) \times  ( 7 \times 7 ) \times ( 12 \times 4 ) \times 47 \times ( 23 \times 2 ) } { 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1  }  \]

\[  52  C _ 7 = \frac  { 26  \times 17 \times 10  \times  7  \times 12 \times 47 \times ( 23 \times 2 ) } {  6 \times 1  }  \]

\[  52  C _ 7 = \frac  { 26  \times 17 \times 10  \times  7  \times 12 \times 47 \times 23  } {  3 \times 1  }  \]

\[  52  C _ 7 = 133,784,560 \]

Numerical Results

The number of ways in which 7 card hands can be chosen from a standard 52-card deck is $ 133,784,560 $.

Example

Find the number of ways the 5-card hands can be chosen from a standard 52-card deck.

According to the combination formula:

\[ 52 C _ 5 = C (  n  ,  r  )  =  C (  52  ,  5 ) \]

\[ 52 C  _ 5 =  \frac  {  52 !  }  {  7 !  (  52  –  7 ) ! } \]

\[  52  C _ 5 = \frac { 52 ! } { 7 ! \times   45 !  }  \]

\[  52  C _ 5 = \frac  { 52 \times 51 \times 50 \times 49 \times 48 \times 47 \times 46 \times 45 ! } { 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \times   45 !  }  \]

\[  52  C _ 5 = \frac  { ( 26 \times 2 ) \times ( 17 \times 3 ) \times ( 10 \times 5 ) \times  49  \times ( 12 \times 4 ) } { 5 \times 4 \times 3 \times 2 \times 1  }  \]

\[  52  C _ 5 = \frac  { 26  \times 17 \times 10  \times  49  \times 12  }  { 1 } \]

\[  52  C _ 5 = 2, 598, 960  \]

The number of ways in which 5 hand cards are arranged is $ 2, 598, 960 $.

Image/Mathematical drawings are created in Geogebra.

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