Find the number of ways in which a selection of $100$ balls, can be made out of $100$ identical red balls, $100$ identical blue balls & $100$ identical white balls

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Solution

Let the number of red, blue and white balls be $x_{1}, x_{2}$ and $x_{3}$ respectively
Therefore, we have $x_{1} + x_{2} + x_{3} = 100$
As we know that if we have
$x_{1} + x_{2} + . . . . . . . . . . . x_{n} = m$
Then no. of ways $=^{m + n - 1} C_{n - 1}$
$∴$ No of ways $=^{100 + 3 - 1} C_{3 - 1} =^{102} C_{2} = 5151$

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Find the number of ways in which a selection of 100 balls, can be made out of 100 identical red balls, 100 identical blue balls & 100 identical white balls

Let the number of red, blue and white balls be x1,x2 and x3 respectivelyTherefore, we have x1+x2+x3=100As we know that if we have x1+x2+...........xn=mThen no. of ways =m+n−1Cn−1∴ No of ways =100+3−1C3−1=102C2=5151

Find the number of ways in which a selection of $100$ balls, can be made out of $100$ identical red balls, $100$ identical blue balls & $100$ identical white balls