What is the probability that in an 'm' digit number, n, less than m, of the digits are repeated
1. Atleast 'k' times
2. at most 'k' times
3. Exactly 'k' times
where k is a number less that m

I'm assuming that the number 'm' is an idealised natural number and so the probability distribution of each digit is from the General Significant-Digit Law derived in Theodore Hill's paper The Significant-Digit Phenomenon. The American Mathematical Monthly 102, 322-327 (1995). This distribution is:

_m_
\ m-i
p(D1,D2 ... Dm) = log10(1 + \ di x 10 )
/
__

Are you assuming that the number 'm' is an idealised natural number and so the probability distribution of each digit is from the General Significant-Digit Law derived in T Hill. The Significant-Digit Phenomenon. The American Mathematical Monthly 102, 322-327 (1995). This distribution explains Benford's Law and is:

_m_
\ m-i
p(D1,D2 ... Dm) = log10(1 + \ di x 10 )
/
/___
i=1

The General Sigificant-Digit Law has the surprising corollary that significant digits are dependent.