In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers?
It is given that out of 17 players, 5 players are bowlers. 4 bowlers are selected in the way that the combination of 5 players taken 4 at a time.
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r ! .
Substitute 5 for n and 4 for r in the above formula.
C 5 4 = 5 ! ( 5 − 4 ) ! 4 ! = 5 ! 1 ! 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 is written as,
C 5 4 = 5 × 4 ! 4 ! = 5
The number of ways that the bowlers are selected is 5.
Now, since 17 players contains 5 bowlers, thus the players left are 12 and similarly, the team of 11 players contain 4 bowlers, thus the players remained are 7.
So, the remaining players are calculated as the combination of 12 players taken 7 at a time.
The formula for the combination is defined as,
C n r = n ! ( n − r ) ! r !
Substitute 12 for n and 7 for r in the above formula.
C 12 7 = 12 ! ( 12 − 7 ) ! 7 ! = 12 ! 5 ! 7 !
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 12 7 = 12 × 11 × 10 × 9 × 8 × 7 ! 7 ! 5 × 4 × 3 × 2 × 1 = 12 × 11 × 10 × 9 × 8 5 × 4 × 3 × 2 × 1 = 792
Thus, the number of ways that the remaining players are selected is 792.
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 cricket team is selected is,
792 × 5 = 3960
Thus, the cricket team can be selected in 3960 ways.
It is given that out of 17 players, 5 players are bowlers. 4 bowlers are selected in the way that the combination of 5 players taken 4 at a time.
The formula for the combination is defined as,
Substitute 5 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 is written as,
The number of ways that the bowlers are selected is 5.
Now, since 17 players contains 5 bowlers, thus the players left are 12 and similarly, the team of 11 players contain 4 bowlers, thus the players remained are 7.
So, the remaining players are calculated as the combination of 12 players taken 7 at a time.
The formula for the combination is defined as,
Substitute 12 for n and 7 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,
Thus, the number of ways that the remaining players are selected is 792.
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 cricket team is selected is,
Thus, the cricket team can be selected in 3960 ways.
