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Intern  G
Joined: 21 May 2016
Posts: 27
During a war a Major is supposed to send 5 units of soldiers to 5 diff  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 31% (02:06) correct 69% (02:13) wrong based on 43 sessions

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During a war a Major is supposed to send 5 units of soldiers to 5 different borders. If he has 30 soldiers and has to send an equal number of soldiers to 5 regions, in how many ways can he do so?

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

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

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

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

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

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Math Expert V
Joined: 02 Aug 2009
Posts: 7957
Re: During a war a Major is supposed to send 5 units of soldiers to 5 diff  [#permalink]

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a70 wrote:
During a war a Major is supposed to send 5 units of soldiers to 5 different borders. If he has 30 soldiers and has to send an equal number of soldiers to 5 regions, in how many ways can he do so?

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

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

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

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

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

Straight formula ..
If you have to make 5 groups of 6 people from (5*6) people, answer is $$\frac{(5*6)!}{(6!)^5}$$

Now if you do not know the formula..
First 6 can be selected in 30C6
Next 6 from remaining 24 in 24C6 and so on
So 30C6*24C6*18C6*12C6*6C6
$$\frac{30!}{24!6!}*\frac{24!}{18!6!}*\frac{18!}{12!6!}*\frac{12!}{6!6!}*\frac{6!}{0!6!}=\frac{30!}{6!6!6!6!6!}=\frac{30!}{(6!)^5}$$

C
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Director  P
Joined: 14 Dec 2017
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Re: During a war a Major is supposed to send 5 units of soldiers to 5 diff  [#permalink]

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a70 wrote:
During a war a Major is supposed to send 5 units of soldiers to 5 different borders. If he has 30 soldiers and has to send an equal number of soldiers to 5 regions, in how many ways can he do so?

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

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

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

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

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

Using the concept, if 2n different items are to divided into 2 groups of n items each, then it can be done in $$\frac{(2n)!}{{n!*n!*2!}}$$

Here we have 30 soldiers, hence 5*6 items to be divided into 5 groups of 6 items each.

30 soldiers can be divided into 5 units of 6 soldiers each in $$\frac{30!}{((6!)^{5}*5!)}$$

The 5 units can then be sent to the 5 different borders in 5! ways.

Hence total # of ways of doing this is = $$\frac{30!}{((6!)^{5}*5!)}*5!$$ = $$30!/(6!)^5$$

Thanks,
GyM
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During a war a Major is supposed to send 5 units of soldiers to 5 diff  [#permalink]

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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.
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Our high-level "quant" preparation starts here: https://gmath.net During a war a Major is supposed to send 5 units of soldiers to 5 diff   [#permalink] 30 Sep 2018, 09:42
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