  We show that for any semi-random transposition shuffle on $n$ cards, the
mixing time of any given $k$ cards is at most $n\log k$, provided $k=o((n/\log
n)^{1/2})$. In the case of the top-to-random transposition shuffle we show that
there is cutoff at this time with a window of size O(n), provided further that
$k\to\infty$ as $n\to\infty$ (and no cutoff otherwise). For the
random-to-random transposition shuffle we show cutoff at time $(1/2)n\log k$
for the same conditions on $k$. Finally, we analyse the cyclic-to-random
transposition shuffle and show partial mixing occurs at time $\le\alpha n\log
k$ for some $\alpha$ just larger than 1/2. We prove these results by relating
the mixing time of $k$ cards to the mixing of one card. Our results rely
heavily on coupling arguments to bound the total variation distance.
