Question 1202169
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9C2*17C3 = 24480 is way too big compared to the upper bound 19C5 = 11628


11628 is the highest we can go.
This is one piece of evidence to allow us to rule out 9C2*17C3


The 2nd calculation you wrote is correct. The answer is <font color=red size=4>9486</font>


9C2 * 10C3 = number of ways to get 2 blue and 3 non-blue
9C3 * 10C2 = number of ways to get 3 blue and 2 non-blue
9C4 * 10C1 = number of ways to get 4 blue and 1 non-blue
9C5 * 10C0 = number of ways to get 5 blue and 0 non-blue



Each row is of the form (mCn)*(pCq)
we have these properties
m+p = 9 blue + 10 nonblue = 19 marbles total
n+q = 5 selections allowed


Calculator verification using WolframAlpha
<a href = "https://www.wolframalpha.com/input?i=%28%289choose2%29+*+%2810choose3%29%29+%2B+%28%289choose3%29+*+%2810choose2%29%29+%2B+%28%289choose4%29+*+%2810choose1%29%29+%2B+%28%289choose5%29%29">https://www.wolframalpha.com/input?i=%28%289choose2%29+*+%2810choose3%29%29+%2B+%28%289choose3%29+*+%2810choose2%29%29+%2B+%28%289choose4%29+*+%2810choose1%29%29+%2B+%28%289choose5%29%29</a>
The input was "((9choose2) * (10choose3)) + ((9choose3) * (10choose2)) + ((9choose4) * (10choose1)) + ((9choose5))" without quotes.


The "choose" function is another name for the nCr combination function.


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Another approach:


There are 19C5 = 11628 ways to pick 5 marbles of any color. Order doesn't matter.


Also we have 10C5 = 252 ways to pick five non-blue marbles
And (9C1)*(10C4) = 9*210 = 1890 ways to pick one blue marble and four non-blue marbles.


Therefore, we have 11628 - (252+1890) = <font color=red>9486</font> ways to select at least two blue marbles in a batch of five.


I'm using the idea that
<font color=blue>n(0 blue) + n(1 blue)</font> + <font color=red>n(at least 2 blue)</font> = n(total)
which leads to
<font color=red>n(at least 2 blue)</font> = n(total) - ( <font color=blue>n(0 blue) + n(1 blue)</font> )
<font color=red>n(at least 2 blue)</font> = 19C5 - ( <font color=blue>10C5 + 9C1*10C4</font> )
<font color=red>n(at least 2 blue)</font> = 11628 - ( <font color=blue>252 + 1890 </font> )
<font color=red>n(at least 2 blue)</font> = <font color=red>9486</font>


Calculator verification using WolframAlpha
<a href = "https://www.wolframalpha.com/input?i=%2819+choose+5%29+-+%2810+choose+5%29+-+%289+choose+1%29*%2810+choose+4%29">https://www.wolframalpha.com/input?i=%2819+choose+5%29+-+%2810+choose+5%29+-+%289+choose+1%29*%2810+choose+4%29</a>
The input was: (19 choose 5) - (10 choose 5) - (9 choose 1)*(10 choose 4)



Another relevant calculator
<a href = "https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html">https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html</a>
It not only computes the nCr value, but it also lists out the various combinations in the "List Them" section.
I recommend exploring with small sets because listing tens of thousands of items would be a tedious task to verify.
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