Exam tmr!!! Please help!!!!! (1 Viewer)

[coeyz]

There are 3 Red balls, 2 Black balls, 2 Yellow balls,
In how many ways that do not have a Red ball at the end of the row?

my way:

B _ _ _ _ _ Y OR Y_ _ _ _ _ B --> 2!5! / 3!
B1_ _ _ _ _ B2 OR B2_ _ _ _ _B1 --> 2!5! / 3! 2!
Y1_ _ _ _ _ Y2 OR Y2_ _ _ _ _Y1 --> 2!5! / 3! 2!

my ans will be 80

but the corrent ans is 60!
how come?? where did i do wrong
thanksss

 

[Drongoski]

There are 3 Red balls, 2 Black balls, 2 Yellow balls,
In how many ways that do not have a Red ball at the end of the row?

my way:

B _ _ _ _ _ Y OR Y_ _ _ _ _ B --> 2!5! / 3!
B1_ _ _ _ _ B2 OR B2_ _ _ _ _B1 --> 2!5! / 3! 2!
Y1_ _ _ _ _ Y2 OR Y2_ _ _ _ _Y1 --> 2!5! / 3! 2!

my ans will be 80
but the corrent ans is 60!
how come?? where did i do wrong
thanksss

Comment: Unless you have a very lucid mind problems in combinatorics can easily make fools of us.

Here's my attempt !

Total number of ways without restriction = 7!/(3! 2! 2!)

No of ways with both ends Red = 5!/(2! 2! 1!)

No of ways left end Red, but right end not Red = 2 x 5!/(2! 2! 1!)
because we fill right end with B or Y (2 ways) and the remaining (2 + 2 + 1)
can be arranged in 5!/(2! 2! 1!) ways.

Similarly no of ways right end Red but not left end = 2 x 5!/(2! 2! 1!)

Therefore no of ways one or both ends not Red = 7!/(3! 2! 2!) - 5 x 5!/(2! 2! 1!)

= (7! - 5 x 3! x 5!)/(3! 2! 2!)

= 60 (without the '!')


Does that make sense ??


PS: Maybe the last line of question itself is ambiguous (as day to day English can be)
Does question mean a Red completely excluded from both ends or do we include, as in my
'solution' where you have a R at one end and not the other.

 

[Timothy.Siu]

There are 3 Red balls, 2 Black balls, 2 Yellow balls,
In how many ways that do not have a Red ball at the end of the row?

my way:

B _ _ _ _ _ Y OR Y_ _ _ _ _ B --> 2!5! / 3!
B1_ _ _ _ _ B2 OR B2_ _ _ _ _B1 --> 2!5! / 3! 2!
Y1_ _ _ _ _ Y2 OR Y2_ _ _ _ _Y1 --> 2!5! / 3! 2!

my ans will be 80

but the corrent ans is 60!
how come?? where did i do wrong
thanksss

ur method is correct, theres no two there because they are the same, blue____blue is the same as blue_____blue right?
so this would make it 5!/3!2!=10
10+10+40=60

 

[coeyz]

Thanks guys~~~