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Fifteen coupons are marked $1,2,3..15$ . Seven coupons are selected at random without replacement . What is the probability that largest number on the selected coupon is 15 ?

My solution :

So total number of sample point is $={{15} \choose {7}} $.

The number of way largest number 15 can be chosen = number of ways choosing 6 other coupons with number $1,2..14 $ .

So total number of favorable cases $= {{14} \choose {6}} $

Hence probability $=\frac {{{14} \choose {6}}}{{{15} \choose {7}}}$

I would like to verify if my reasoning is correct . Please tell if i'm missing out some cases or i'm double counting .

Thank you .

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Yes. Your solution is correct. If we didn't care about order of coupons.

One largest number coupon is selected. Then remaining 6 cards are selected from remaining 14 coupons.

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  • $\begingroup$ Thank you . I would also like to know if the the condition " without replacement " is replaced by "with replacement " then answer would be $ \frac {14^6}{15^7} $ . Is this correct ? $\endgroup$ – Suman Kundu Apr 19 '17 at 16:36
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    $\begingroup$ If with replacement you have case where all cards can be largest. $\endgroup$ – Kanwaljit Singh Apr 19 '17 at 16:40

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