In how many ways can 4 identical red balls, 6 identical blue balls and

Question

In how many ways can 4 identical red balls, 6 identical blue balls and 8 identical yellow balls be arranged in a row so that at least one ball is separated from the balls of the same color?

A.  3!

B.  \frac{18!}{4!6!8!}-6

C.  \frac{18!}{4!6!8!}-3

D.  \frac{18!}{4!6!8!}-1

E.  \frac{18!}{4!6!8!}

HOW MANY WAYS CAN 4 IDENTICAL RED BALLS AND 6 IDENTICAL BLUE BALLS AND 8 IDENTICAL YELLOW
be arranged in a row so that at least one is separated from balls of the same color

ThatDudeKnows · Answer

Answer choice A is just obviously wrong. The other answer choices are all identical except for what we are subtracting from the factorial fraction fun, so we can just ignore the factorial fraction fun and figure out what's being subtracted. It must be the number of ways to arrange the balls in a manner that violates the rules. The only ways to do that are to put all the balls of one color first followed by all the balls of a second color followed by all the balls of the third color. The only ways to do that are:

all the red then all the blue then all the yellow
all the red then all the yellow then all the blue
all the blue then all the red then all the yellow
all the blue then all the yellow then all the red
all the yellow then all the red then all the blue
all the yellow then all the blue then all the red

So that's six ways that we need to subtract.

Answer choice B.

anon7164 · Answer

Number of ways that at least one ball is separated from the balls of the same colour = Total ways of arranging the balls - number of ways that all balls are together.

Total ways: 18!/4!6!8! (think RRRRBBBBBBYYYYYYYY)
Number of ways of all balls being together: we have 3 types (colours) of balls. Since we want all balls of the same colour to be together, divide the balls into 3 groups, with each group composed of only a single colour. Number of ways of arranging these 3 groups = 3!
(We do not have to worry about number of ways of arranging the balls within their respective groups, because balls of the same colour are identical.)

Therefore, B.

Varane · Answer

Can you elaborate why we subtract 3! From the total ways.

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manman3402 · Answer

As the balls are identical, we can consider them as 1 block. and there are 3 types of ball. so we caculate total ways as 3!/1!x1!x1! which is 3!. And then to find the, we subtract the total possibilities - 3!.

In how many ways can 4 identical red balls, 6 identical blue balls and

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In how many ways can 4 identical red balls, 6 identical blue balls and

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