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you can have

5 engineers and 10 professors in C(10,5)*C(12,10) =16632 ways

6 " and 9 " in C(10,6)*C(12,9)=46200 ways

7 " and 8 " in C(10,7)*C(12,8)=59400 ways

8 " and 7 " in C(10,8)*C(12,7)=35640 ways

9 " and 6 " in C(10,9)*C(12,6)=9240 ways

10 " and 5 " in C (10,10)*C(12,5)=792 ways

therefore different committees can be formed in (16632+46200+59400+35640+9240+792)=167904 ways

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