Question 1196897: Bailey the dog owns 8 balls. 4 are colored red, 3 are colored yellow, and 1 is blue. If she fetches 3 balls from her collection, how many different color combinations are possible?
Found 3 solutions by ikleyn, math_tutor2020, greenestamps:
Answer by ikleyn(52776)   (Show Source):
You can put this solution on YOUR website! .
The problem asks about the number of different color combinations.
It means that we distinct the balls only due to their colors -
- the balls with identical color are indistinguishable.
So, the math model is the set of all possible 3-letter words written
using the alphabet consisting of 3 letter
R (symbolizing red color),
Y ( yellow color) and
B ( blue color)
with the restriction that the letter B can be used not more than once in each word.
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| First consider the case when B is used exactly once in the words. |
+---------------------------------------------------------------------+
If the order of colors is important, then the number of such words
(different color combinations) with one B is 3*X*Z,
where X can be any of R or Y, without restrictions
and Z can be any of R or Y, without restrictions.
The factor 3 symbolize that "B" may stand in any of three position in the word.
So, the number of such words (different color combinations) with one B is 3*2*2 = 12,
if the order of colors is important.
If the order of colors is not important, then all possible color combinations with one B are
BRR, BRY, BYY, giving only 3 combinations.
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| Next consider the case when B is NOT used in the words. |
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Then, if the order of colors is important, we can have any of two remaining colors
in each position, giving 2*3 = 6 different color combinations.
If the order of colors is not important, then we have these combinations
RRR, RRY, RYY, YYY, giving 4 possible color combinations.
ANSWER. The number of all different color combinations is
12 + 6 = 18, if the order is important;
3 + 4 = 7, if the order is not important.
Solved.
Answer by math_tutor2020(3816)  (Show Source):
You can put this solution on YOUR website!
Answer: 7
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Reasoning:
Let,
B = blue
R = red
Y = yellow
The given data set is
B,RRRR,YYY
the task is to select three of these to see how many color combos we can form.
The seven different color combos are:- BRR
- BRY
- BYY
- RRY
- RYY
- RRR
- YYY
Here's the process I did:
Start with B which is toward the beginning of the alphabet compared to R and Y. There's only one ball of this color so we can't select another B.
Then select the next highest letter R, followed by another R
That forms BRR meaning a blue ball, red, and then another red is selected.
The order doesn't matter due to the word "combinations" (in contrast to permutations where order does matter)
Then place the balls back.
Select B again, then R, then Y this time
That forms BRY which is the second item on the list.
The idea is to sequentially scroll through the terms alphabetically until fully exhausting the list of all seven possibilities.
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
You have two responses showing very different paths of reasoning to reach the answer of 7.
Here is a different path that I personally find quite a bit easier....
(1) 3 of the same color: RRR or YYY -- 2 combinations
(2) 2 of one color and one of another: RRY or RRB; or YYR or YYB -- 4 combinations
(3) 1 of each color: RYB -- 1 combination
ANSWER: 2+4+1 = 7
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