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 .