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How many different $3-$digit numbers are there with the sum of digits as $15$ ? 
Overseas Competition->Knowledge Point->Counting Modules
To divide $15+1+1=17$ into three groups, there are $\_{16}C\_{2}=120$ ways in total.  But, since all digits cannot be larger than $9$, except the situations that don\textquotesingle t meet the requirements, there are $120-\_7C\_2-\_6C\_2\times 2=69$ numbers that match the condition. 