Hi!
What is the probabilty that in a 10-digit number, digit 7 repeats 6 times (no more or less, exactly)?
or in other words, what is the probability that in a random 10-digit number there will be 6 sevens.
thanks
First by "10-digit number' you must mean the leading digit cannot be 0.
So there are $\displaystyle 9\cdot 10^9$ 10-digit numbers.
Of those, there are $\displaystyle \binom{9}{5}\cdot 9^4$ that begin with a seven and have exactly five other sevens.
There are $\displaystyle 8\cdot\binom{9}{6}\cdot 9^3$ that do not begin with a seven and have exactly six sevens.