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$15$ coupons are numbered $1,2,3...,15$ respectively. $7$ coupons are selected at random one at a time with replacement. The probability that the largest number appearing in a selected coupon is $9$ ???

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The largest number will be $9$ if out of $7$ selected coupon one is numbered $9$ and the other $6$ are in between $1$ to $8$.The number $9$ can be choosen in any one place out the $7$ selected coupons. So there are $7$ ways I can choose $9$ and in each case other $6$ numbers can be choosen in $8^6$ ways. Hence the required probability is $\frac {7\times 8^6}{{15}^7}$
I know exactly the same problem is there in Stack. But my approach is completely different. And I want to know where is the mistake in this method.

RobPratt
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Math-Learner
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    Let $M$ denote the max of the coupons. The probability that all are less than $i$ is $P(M≤i)=\left(\frac i{15}\right)^7$. Can you finish from here? – lulu Apr 07 '22 at 16:19
  • It seems you neglected the probability of selecting the $9$ exactly, which is $1/15.$ – William M. Apr 07 '22 at 16:20
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    There could be multiple $9's$ which you have not accounted for – true blue anil Apr 07 '22 at 16:21
  • Indeed, you want there to be $1$ nine, and $6$ that are less or equal than nine. The problem will become messy because you no longer will have control on where the $9$ falls. A better approach is to use @lulu's – William M. Apr 07 '22 at 16:23

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