Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.
There are 4 aces in a deck of 52 cards. As there should be exactly one ace in each combination of 5 cards, thus one ace can be selected as a combination of 4 aces taken 1 at a time.
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r !
Substitute 4 for n and 1 for r in the above formula.
C 4 1 = 4 ! ( 4 − 1 ) ! 1 ! = 4 ! 3 ! 1 !
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
n ! = n ( n − 1 ) ! n ! = n ( n − 1 ) ( n − 2 ) ! [ n ≥ 2 ] ⋮
The combination is written as,
C 4 1 = 4 × 3 ! 3 ! = 4
The number of ways that the aces are selected is 4.
Since in the combination of 5 cards, one place is occupied by an ace, thus there remain 4 cards and also the total number of cards left is 48 after the removal of 4 aces from 52 cards. So, the number of ways of selecting the cards is the combination of 48 cards taken 4 at a time.
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 48 for n and 4 for r in the above formula.
C 48 4 = 48 ! ( 48 − 4 ) ! 4 ! = 48 ! 44 ! 4 !
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
n ! = n ( n − 1 ) ! n ! = n ( n − 1 ) ( n − 2 ) ! [ n ≥ 2 ] ⋮
The combination can be written as,
C 48 4 = 48 × 47 × 46 × 45 × 44 ! 44 ! 4 × 3 × 2 × 1 = 48 × 47 × 46 × 45 4 × 3 × 2 × 1 = 194580
The number of ways that the cards are selected is 194580.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the total number of ways is m × n .
The number of ways that the 5 card combinations are made is,
194580 × 4 = 778320
Thus, required number of 5 card combinations is 778320.
There are 4 aces in a deck of 52 cards. As there should be exactly one ace in each combination of 5 cards, thus one ace can be selected as a combination of 4 aces taken 1 at a time.
The formula for the combination is defined as,
Substitute 4 for n and 1 for r in the above formula.
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
The combination is written as,
The number of ways that the aces are selected is 4.
Since in the combination of 5 cards, one place is occupied by an ace, thus there remain 4 cards and also the total number of cards left is 48 after the removal of 4 aces from 52 cards. So, the number of ways of selecting the cards is the combination of 48 cards taken 4 at a time.
The formula for the combination is defined as,
Substitute 48 for n and 4 for r in the above formula.
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
The combination can be written as,
The number of ways that the cards are selected is 194580.
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, the total number of ways is
The number of ways that the 5 card combinations are made is,
Thus, required number of 5 card combinations is 778320.
