Combination problems

David99

New member
Joined
May 16, 2019
Messages
2
1. In how many ways can 4 red balls, 3 white balls and 1 black ball be arranged in a line so that the black ball is always surrounded by a red and a white ball?
2. In how many ways can the balls be arranged if no red ball is next to a white?
Thank you in advanced.
 

Dr.Peterson

Elite Member
Joined
Nov 12, 2017
Messages
3,650
Please give it a try and show what you have done. That will give us a place to start from in helping you. Keep in mind that our purpose is to help you learn to solve the problem, not just to do it ourselves and show off our skill.

My first thought for (1) is to first choose a place for the black, then choose the order of the two balls around it, and then place the rest of the balls.
 

pka

Elite Member
Joined
Jan 29, 2005
Messages
8,128
1. In how many ways can 4 red balls, 3 white balls and 1 black ball be arranged in a line so that the black ball is always surrounded by a red and a white ball?
2. In how many ways can the balls be arranged if no red ball is next to a white?
Thank you in advanced.
For 1) 'glue a red ball to the black ball then 'glue' a white to the other side of the black ball.
How many ways can the block of three along with the remaining three red balls & two white balls (six units in all) be arranged?

Now for 2) here are two correct strings:
\(\displaystyle r\,r\,r\,r\,b\,w\,w\,w\text{ Or }w\,w\,w\,b\,r\,r\,r\,r\)
Is there any other possible strings?
 
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