jedit wrote:
A bibliophile plans to put a total of seven books on her marble shelf. She can choose these seven books from a mixture of works from Antiquity and works on Post-modernism, of which there are seven each. If the shelf must contain at least four works from Antiquity, and one on Post-Modernism, then how many ways can he select seven books to go on the shelf?
A - 441
B - 1225
C - 1666
D - 1715
E - 1820
I solved the same way as others, and just want to highlight a time saver: At each stage, this question involves the identity
\(_nC_k = {_n}C_{(n-k)}\)
In other words,
\(_7 C _4 = {_7}C _3\)
\(_7 C _5 = {_7}C _2\)
\(_7 C _6 = {_7}C _1\)
Basically, selecting which \(k\) objects DO get chosen is the same as selecting which \((n - k)\) objects DO NOT get chosen.
Once you compute the first combination in each pair, you're done; it's exactly the same value as the second combination in the pair.
The point is obvious for people who are really fluent in combinations. I'm not. I don't like them.
So knowing that identity saved me a decent chunk of time here. Cheers!
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