**Quote:**

During a war a Major is supposed to send 5 units of 6 soldiers in each unit to 5 different borders. Into how many groups the 30 soldiers may be divided?

A. \(30!/(5!)^6\)

B. \(30!/(5!)^5\)

C. \(30!/(6!)^5\)

D. \((30!/6!^5)5!\)

E. \(30!/(6!^5*5!)\)

[In the original question stem, we could send (say) just 5 soldiers -one to each border - from the 30 soldiers available, without violating the restrictions imposed.]

\(?\,\,:\,\,\,\# \,\,{\rm{groups}}\)

Border A: C(30,6) choices

Border B: C(24,6) choices

Border C: C(18,6) choices

Border D: C(12, 6) choices

Border E: C(6,6) choices or, if you prefer, the soldiers left ("no choices" then).

Using the

Multiplicative Principle, we have:

\(?\,\,\, = \,\,\,\left( {{{30!} \over {6!\,\,24!}}} \right)\,\,\left( {{{24!} \over {6!\,\,18!}}} \right)\left( {{{18!} \over {6!\,\,12!}}} \right)\,\,\left( {{{12!} \over {6!\,\,6!}}} \right)\,\,\, = \,\,\,{{30!} \over {{{\left( {6!} \right)}^5}}}\)

This solution follows the notations and rationale taught in the GMATH method.

Regards,

fskilnik.

_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)

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