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In the following problem:

Noor has 3 T-shirts, 4 blouses and 5 jumpers. She chooses three randomly. What is the probability she chooses exactly one T-shirt?

The probability for the event pick one T-shirt and two other tops is

$3/12 * 9/11 * 8/10$

which I would then times by three for the three different ways it could occur ( TOO; OTO and OOT where T= T-shirt and O-other).

However apparently I should times by

$^{3}C_{1}$

I know this also equals three but it suggests that which T-shirt out of the three picked is important. That inturn suggests which of the other items picked is important and complications ensue...

There must be some mistake in my understanding of the situation, what is it?

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    Indeed, $3=3$, so what? You pick the one position out of three positions where the T occurs in the sequence. -- Anyway, I'd rather compute $\frac{{3\choose 1}{4+5\choose 2}}{3+4+5\choose 3}$ – 2017-02-25

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You are including t-shirts in other cases also.

Instead you can do it as -

$\binom 31 \times \binom92$

And then multiply by 3.

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    Any doubt you can ask. – 2017-02-25