Question 1204889
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Answers:
(b) <font color=red>87,178,291,200</font>
(c) <font color=red>72,072</font>



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 = <font color=red>87,178,291,200</font>
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!}}} where k is the number of items in that group.
For the reds we divide by {{{7!}}} 
For the yellows we divide by {{{2!}}} 
For the greens we divide by {{{5!}}} 


So,
{{{(14!)/(7!*2!*5!)}}}


{{{(14*13*12*11*10*9*8*7*6*5*4*3*2*1)/((7*6*5*4*3*2*1)*(2*1)*(5*4*3*2*1))}}}


{{{(14*13*12*11*10*9*8*highlight(7*6*5*4*3*2*1))/((highlight(7*6*5*4*3*2*1))*(2*1)*(5*4*3*2*1))}}}


{{{(14*13*12*11*10*9*8*cross(7*6*5*4*3*2*1))/((cross(7*6*5*4*3*2*1))*(2*1)*(5*4*3*2*1))}}}


{{{(14*13*12*11*10*9*8)/((2*1)*(5*4*3*2*1))}}}


{{{(cross(14)^7*13*12*11*10*9*8)/(cross(2)*1*5*4*3*2*1)}}}


{{{(7*13*12*11*10*9*8)/(5*4*3*2*1)}}}


{{{(7*13*12*11*cross(10)^2*9*8)/(cross(5)*4*3*2*1)}}}


{{{(7*13*12*11*2*9*8)/(4*3*2*1)}}}


{{{(7*13*cross(12)^1*11*2*9*8)/(cross(4)*cross(3)*2*1)}}}


{{{(7*13*1*11*2*9*8)/(2*1)}}}


{{{(7*13*1*11*cross(2)*9*8)/(cross(2)*1)}}}


{{{(7*13*1*11*1*9*8)/(1)}}}


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


In short,
{{{(14!)/(7!*2!*5!) = 72072}}}
There are <font color=red>72,072</font> marble arrangements possible if we cannot distinguish the same color marbles from one another.
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