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In how many different ways it is possible to seat 5 women and 5 men in 10 seat spots available where all 5 women seat next to each other?

I'm thinking $6\cdot5!\cdot5!$

6-different spots between the men

5!- for men

5!- for women

Is that correct? Thanks.

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Do all the women have to be sitting next to each other? Or does each one just need to be sitting next to another? – Ataraxia Aug 21 '13 at 18:21
all women have to seat next to each other. I edited my question, sorry for that. – Mike L Aug 21 '13 at 18:28
it means all together right ? – Harish Kayarohanam Aug 21 '13 at 18:29
Yes, your analysis is correct. See @Harish’s answer for another way of arriving at the same conclusion. – Brian M. Scott Aug 21 '13 at 18:29
up vote 3 down vote accepted

all women sitting next to each other I hope means all together.Then you are correct.

I will solve like

5 women considered as a unit . so 5 men + 1 women unit .arrange this first . then arrange women among themselves

$ 6! \times 5! $

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If all women need to be sitting next to each other, you're right. As a sanity check, you can check if your answer is less than 10! (it is). This doesn't prove that it's right, of course, but if it were bigger it would certainly be wrong.

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Another quick question, Is it possible to feel that $6!\cdot5!$ is less then 10!? (without calculator) – Mike L Aug 21 '13 at 18:32
6! * 5! = 6! * (5*2) *(4*3)= 6! * 10 * 12. But 10! is 6! * 10 * (9*8*7) . even 9*8 = 72 is greater than 12 so 10! > 6!*5! – Harish Kayarohanam Aug 21 '13 at 18:37
I don't think you can "feel" it without at least thinking a bit about it as Harish did. – MGA Aug 21 '13 at 18:41

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