# Combination question?

• Jul 9th 2011, 07:45 AM
wabbt
Combination question?
A jar contains ten blue balls, ten red ones, ten white ones, and ten green ones. Each set of ten balls of the same colour is numbered from 1 to 10. How many different selections of four balls are possible if exactly three of the balls bear the same numbers?

The answer's supposed to be 1440.

Can someone show me the steps of solving this problem? Thanks!
• Jul 9th 2011, 08:48 AM
abhishekkgp
Re: Combination question?
Quote:

Originally Posted by wabbt
A jar contains ten blue balls, ten red ones, ten white ones, and ten green ones. Each set of ten balls of the same colour is numbered from 1 to 10. How many different selections of four balls are possible if exactly three of the balls bear the same numbers?

The answer's supposed to be 1440.

Can someone show me the steps of solving this problem? Thanks!

CASE (i)
suppose the three balls bearing the same number bear the number 1.
there are four subcases:
1) those three balls are B,R,G
2) those three balls are B,R,W
3) those three balls are R,G,W
4) those three balls are B,G,W

for subcase 1) there are 36 possible choices. (why?)

So for CASE(i) there are 4*36 choices.

Hence the total number of choices are 4*10*36.
• Jul 9th 2011, 08:55 AM
Plato
Re: Combination question?
Quote:

Originally Posted by wabbt
A jar contains ten blue balls, ten red ones, ten white ones, and ten green ones. Each set of ten balls of the same colour is numbered from 1 to 10. How many different selections of four balls are possible if exactly three of the balls bear the same numbers? The answer's supposed to be 1440.

There ten ways to pick the number that occurs three times.
The there are thirty-six ways we can pick the odd-man out.
There are four ways each of those selections can be made.
• Jul 9th 2011, 09:21 AM
wabbt
Re: Combination question?
How come it's 36 instead of 37? Since we only need 3 balls of the same color and there would be 37 left to choose from?
• Jul 9th 2011, 09:50 AM
Plato
Re: Combination question?
Quote:

Originally Posted by wabbt
How come it's 36 instead of 37? Since we only need 3 balls of the same color and there would be 37 left to choose from?

Once we have the number to be used three times in effect we have ruled out four balls because we cannot the forth ball of that number. That leaves thirty-six to choose.
• Jul 9th 2011, 01:17 PM
Soroban
Re: Combination question?
Hello, wabbt!

Quote:

A jar contains ten blue, ten red, ten white, and ten green balls.
Each set of ten balls of the same colour is numbered from 1 to 10.
How many different selections of four balls are possible
if exactly three of the balls bear the same numbers?

The answer's supposed to be 1440.

Consider this analogous problem . . .

Remove the face cards from a standard deck of cards.
You 40 cards, from Ace to Ten in four suits.
You draw four cards.
In how many ways can you get Three-of-a-Kind?

The are 10 choices for the value of the triple.
There are $\displaystyle {4\choose3} = 4$ ways to get the triple.
There are 36 choices for the fourth card.
. . (We do not want Four-of-a-Kind.)

Therefore, there are: .$\displaystyle 10\cdot4\cdot 36 \:=\:1440$ ways.