I solved a problem, by two different, and (apparently) viable solutions. But, each solution led me to a different answer:(adsbygoogle = window.adsbygoogle || []).push({});

The question asks me to find out the number of possible ways in which I can choose 5 cards from a standard pack of 52 cards, so that I choose at least a single card from each suite.

FIRST METHOD:

One card from each suite of 13 means a total of [^{13}C_{1}*^{13}C_{1}*^{13}C_{1}*^{13}C_{1}].

After that, I'll have 52-4=48 cards left from which I need to choose only one. That means^{48}C_{1}.

The final result:

^{13}C_{1}*^{13}C_{1}*^{13}C_{1}*^{13}C_{1}* 48

SECOND METHOD:

I'll be choosing one card from three suites, but exactly two cards from the fourth suite. So that means:

^{13}C_{1}*^{13}C_{1}*^{13}C_{1}*^{13}C_{2}

=^{13}C_{1}*^{13}C_{1}*^{13}C_{1}*^{13}C_{2}* (12/2)

And since I do this with each of the four suites (choose two from it only, while taking one from each of the others), I shall multiply this by 4, giving me:

^{13}C_{1}*^{13}C_{1}*^{13}C_{1}*^{13}C_{2}* 24

And this is half of the answer arrived at by the first method....

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what went wrong????

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# Combinatronics Dilemma

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