Combinatorics: Select at least one full professor I have a question in the combinatorics section of my book

The Math department of a small college has three full professors, seven associate professors, and four assistant professors. In how many ways can a four member committee be formed under the restriction that at least one full professor is selected.

The way I solved this problem was to see it as at least 1 full professor or at least 2 full professors or at least 3 full professors
$3\choose 1$ $11\choose 3$ $+$ $3\choose 2$ $11\choose 2$ $+$ $3\choose 3$ $11\choose 1$
But the answer in the book is $14\choose 4$ $-$ $11\choose 4$. Can someone tell me why my way of doing it is wrong and why book's answer is right?
 A: Add the number of associate professors and assistant professors $(7 + 4) = 11$ to compute the number of professors who are not full-professors.    
There are:


*

*$3$ full professors

*$11$ professors who are not full-professors.

*$14$ total professors
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There are a few other quantities of interest:    


*

*$\mathrm{ANSWER}$: The number of ways that a four member committee be formed under the restriction that $\underline{\textbf{at least one}}$ full professor is selected.

*$\mathrm{TOTAL}$: The number of ways that a four member committee be formed under no special restrictions

*$\mathrm{UNDESIRABLE}$:The number of ways that a four member committee be formed under the restriction that $\underline{\textbf{no}}$ full professors are on committee at all.
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The author of your textbook was thinking that:
$\mathrm{ANSWER} = \mathrm{TOTAL} - \mathrm{UNDESIRABLE}$
Note that:
$\mathrm{TOTAL} =  \binom{14}{4}$ 
$\mathrm{UNDESIRABLE} =  \binom{11}{4}$
$\mathrm{UNDESIRABLE}$ is $\binom{11}{4}$ because you are choosing 4 people from the pool of 11 non-full professors.
A: As pointed out in the comments, your answer is fine. The book probably reasoned via "inclusion exclusion," i.e. there are $\binom{14}{4}$ total committees that can be created, and of those $\binom{11}{4}$ contain no full professors, thus the desired answer is the difference between the two.
