Question

Consider a Little League team that has 15 players on its roster.

(a)

How many ways are there to select 9 players for the starting lineup?

ways

(b)

How many ways are there to select 9 players for the starting lineup and a batting order for the 9 starters?

ways

(c)

Suppose 7 of the 15 players are left-handed. How many ways are there to select 3 left-handed outfielders and have all 6 other positions occupied by right-handed players?

ways

Answer 1

Answer:

Given,

n = 15

a)

To give the number of ways to select 9 players for starting lineup

Number of ways = 15C9

= 15!/(15-9)!*9!

= 15! / 7!*9!

= (15*14*13*12*11*10*9!) / (7*6*5*4*3*2*1 * 9!)

= 15*14*13*12*11*10 / 7*6*5*4*3*2*1

= 3603600/5040

Number of ways = 715

b)

To give number of ways  to select 9 players for the starting lineup and a batting order for the 9 starters

Here number of ways to select the 9 players for starting lineup = 15C9 * 9!

They can arrange themselves in 9!

So number of ways = 15C9 * 9! *9!

= 659,067,881,472,000

c)

To give the number of ways  to select 3 left-handed outfielders and have all 6 other positions occupied by right-handed players

i.e.,

left hand players = 7

Right hand players = 8

Number of ways to select 3 left handed from 7 = 7!/(7-3)!

= 7!/4!

Number of ways to select 6 right handed players from 8 = 8!/(8-6)!

= 8!/2!

Total number of ways = 7!/4! * 8!/2!

= 210*20160

= 4233600

Total number of ways = 4233600