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# How many positive integers can be expressed as a product of

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Manager
Joined: 28 Apr 2003
Posts: 95
Location: Singapore
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How many positive integers can be expressed as a product of [#permalink]  06 Dec 2003, 21:56
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How many positive integers can be expressed as a product
of two or more of the prime numbers 5, 7, 11, and 13 if no
one product is to include the same prime factor more
than once?

a. 8
b. 9
c. 10
d. 11
e. 12

I could only get the below workings, and I am not sure if someone can provide a shortcut way to derive it.

5x7
5x11
5x13
7x11
7x13
11x13
Director
Joined: 28 Oct 2003
Posts: 503
Location: 55405
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Quote:
How many positive integers can be expressed as a product
of two or more of the prime numbers
(emphasis mine)

As you have discovered, there are 6 (4 choose 2) pairs.
You have neglected sets of three (4 choose 3) 4
and the set of four 1

6+4+1=11
Manager
Joined: 28 Apr 2003
Posts: 95
Location: Singapore
Followers: 1

Kudos [?]: 0 [0], given: 0

5x7x11
5x7x13
5x11x13
7x11x13
5x7x11x13
SVP
Joined: 03 Feb 2003
Posts: 1608
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guys, learn to apply combinatorics: 4C2+4C3+4C4=6+4+1=11
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