Possible combinations of drink orders with 7 orders, 3 sizes. With condition

Question

I have a homework problem that I can't wrap my head around.

You have $7$ drink requests. $3$ possible sizes: sm, med, lg. How many possible ways can you order each drink assuming at least two orders are small?

At first I thought it might be:

$O1 = sm$

$O2 = sm$

$O3 = sm, med, lg$

$O4 = sm, med, lg$

$O5 = sm, med, lg$

$O6 = sm, med, lg$

$O7 = sm, med, lg$

Then I realized there's $6$ ways you could order that assuming the first order is the sm.

Then there are $5$ ways if the second order is sm.

Then $4$ ways if the 3rd order is sm.

I'm getting lost in the possibilities. Can someone help me out please?

Answer 1

Without condition, there are $3^{7}$ possible order combinations. From these, $2^{7}$ does not have small size order and $7\times 2^{6}$ only have one small size order. Thus the answer is $3^{7}-2^{7}-7\times 2^{6}$

Generally, try to avoid listing the possibilities (brute force). If the calculation is hard, try to calculate the complement set, in this case, instead of counting possible orders with at least 2 small size, I opted to count. possible orders without small size and with 1 small size, which is easier and i can subtract these to obtain the answer.