permutation n combination

BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

This topic has expert replies

Post new topic Post Reply

Previous Topic Next Topic

Reva: Senior | Next Rank: 100 Posts; Posts: 93; Joined: Fri Feb 19, 2010 11:52 am; Thanked: 1 times

permutation n combination

by Reva » Thu Mar 11, 2010 3:05 pm

can anyone plz give me the right answer

Question : There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is:

Quote

Source: Beat The GMAT — Problem Solving | Thu Mar 11, 2010 3:05 pm

Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pm; Location: Toronto; Thanked: 1710 times; Followed by:614 members; GMAT Score:800

by Stuart@KaplanGMAT » Thu Mar 11, 2010 11:49 pm

Reva wrote:can anyone plz give me the right answer

Question : There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is:

In future, please provide the choices and the source!

Let's break this up into 6 cases:

1 green ball, 5 red balls: 6 possible places for the green ball.

2 green balls, 4 red balls: the green balls can go in boxes 1/2, 2/3, 3/4, 4/5 or 5/6... that's 5 more possibilities.

3 green balls, 3 red balls: the green balls can go in boxes 1/2/3, 2/3/4, 3/4/5 or 4/5/6... that's 4 possibilities.

4 green balls, 2 red balls: the green balls can go in boxes 1/2/3/4, 2/3/4/5 or 3/4/5/6... that's 3 possibilities.

5 green balls, 1 red ball: the green balls can go in boxes 1/2/3/4/5 or 2/3/4/5/6... that's 2 possibilities.

6 green balls, 0 red balls: only 1 possibility.

So, there are 6+5+4+3+2+1 = 21 possible ways we can meet the requirements.

Of course, at some point (likely after the 2nd or 3rd step) we would have recognized the pattern and saved ourselves some grunt work.