I found this formula in a website: Number of ways in which one or more objects can be selected out of $S_1$ alike objects of one kind, $S_2$ alike objects of second kind and $S_3$ alike objects of third kind $= (S_1 + 1)(S_2 + 1)(S_3 + 1) - 1$.

Where did they get this from? Does this use the concept of multinomial theorem? How is it derived?

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    $\begingroup$ Pick how many objects of the first type you take. You have the options of having taken $0,1,2,3,\dots,S_1$ for a total of $S_1+1$ options. Then pick how many objects of the second type. You will similarly have $S_2+1$ options, and then $S_3+1$ options for the third type. Apply rule of product. Then recognize that one of those results corresponded to having selected none of the first, second, or third types. Since we wanted to have picked one or more, we didn't want to pick only zero, so correct the count. $\endgroup$ – JMoravitz May 4 '18 at 5:33
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    $\begingroup$ No, this has absolutely nothing to do with the multinomial theorem. It is more fundamental, having relied on the rule of product and the rule of sum. $\endgroup$ – JMoravitz May 4 '18 at 5:34
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    $\begingroup$ @JMoravitz Post it as an answer! It makes sense to me! $\endgroup$ – Christopher Marley May 4 '18 at 5:37
  • $\begingroup$ Please read this tutorial on how to typeset mathematics on this site. $\endgroup$ – N. F. Taussig May 4 '18 at 9:12

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