SOLUTION: A boy has 7 red , 2 yellow and 5 green marbles. In how many ways can the boy arrange the marbles in a line if: b) All marbles have different sizes? c) Marbles of the same color a

Algebra ->  Permutations -> SOLUTION: A boy has 7 red , 2 yellow and 5 green marbles. In how many ways can the boy arrange the marbles in a line if: b) All marbles have different sizes? c) Marbles of the same color a      Log On


   

Question 1204889: A boy has 7 red , 2 yellow and 5 green marbles. In how many ways can the boy arrange the marbles in a line if:
b) All marbles have different sizes?
c) Marbles of the same color are indistinguishable?
I'd need both parts solved pls.

Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Answers:
(b) 87,178,291,200
(c) 72,072


Explanation for part (b)
We have 7 red + 2 yellow + 5 green = 14 marbles total.
There are 14 choices for the 1st slot, then 13 choices for the next slot, and so on.
14! = 14*13*12*11*10*9*8*7*6*5*4*3*2*1 = 87,178,291,200
The exclamation mark represents factorial.
This value is a bit over 87 billion.
When converting to scientific notation, we get the approximate value 8.718 * 10^10
Another way to reach this value is to use the nPr permutation formula with n = 14 and r = 14.


Explanation for part (c)
In the previous part, we could tell marbles of the same color apart.
But now we have 7 red we cannot tell apart, and 2 yellow we cannot tell apart, and also 5 green we cannot tell apart.
For each group where we cannot tell them apart, we must divide by the factorial k%21 where k is the number of items in that group.
For the reds we divide by 7%21
For the yellows we divide by 2%21
For the greens we divide by 5%21

So,
%2814%21%29%2F%287%21%2A2%21%2A5%21%29







%2814%2A13%2A12%2A11%2A10%2A9%2A8%29%2F%28%282%2A1%29%2A%285%2A4%2A3%2A2%2A1%29%29



%287%2A13%2A12%2A11%2A10%2A9%2A8%29%2F%285%2A4%2A3%2A2%2A1%29

%287%2A13%2A12%2A11%2Across%2810%29%5E2%2A9%2A8%29%2F%28cross%285%29%2A4%2A3%2A2%2A1%29

%287%2A13%2A12%2A11%2A2%2A9%2A8%29%2F%284%2A3%2A2%2A1%29



%287%2A13%2A1%2A11%2A2%2A9%2A8%29%2F%282%2A1%29

%287%2A13%2A1%2A11%2Across%282%29%2A9%2A8%29%2F%28cross%282%29%2A1%29

%287%2A13%2A1%2A11%2A1%2A9%2A8%29%2F%281%29

72072
Your steps do not need to be as verbose. A calculator can make quick work of this.

In short,
%2814%21%29%2F%287%21%2A2%21%2A5%21%29+=+72072
There are 72,072 marble arrangements possible if we cannot distinguish the same color marbles from one another.