Binomial probability question

#1
Hi all, I have been stumped on this binomial probability question and was wondering if anyone can help me get started:

A hotel has a capacity of 100 rooms. To reduce losses due to cancellations, reservations are accepted in excess of its total capacity. In the past, the motel has found that 20% of prospective guests will cancel their reservation at the last minute.

If the hotel accepts 120 reservations, what is the probability that all the guests who claim a room will receive one (ie: number of guests that show up will not exceed total capacity)
I was thinking X~Bin(100,0.2)
where success=cancellation

and 20% of 120 reservations will cancel, so that is 0.2*120=24
So number of reservations is 120-24=96 with no cancellation is 96

after that I am stuck
 
#2
It should be X~Bin(120,0.2)

We need 20 or more cancellations

\(\sum\limits_{i=20}^{120} \binom{120}{i} * {.2}^{i} * {.8}^{120-i}\)

or

\(1 - \sum\limits_{i=0}^{19} \binom{120}{i} * {.2}^{i} * {.8}^{120-i} \approx 0.1517\)