Question 1150832
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The groups of 3 persons can be formed from 12 persons by  {{{C[12]^3}}} = {{{(12*11*10)/(1*2*3)}}} =  220 ways (helicopter committee).


The groups of 5 persons can be formed from 12 persons by  {{{C[12]^5}}} = {{{(12*11*10*9*8)/(1*2*3*4*5)}}} = 792  ways (Glider committee).


If no other restrictions, then these two committees can be formed by  {{{C[12]^3}}}.{{{C[12]^5}}} = 220*792 = 174240 ways.


From this number, we must subtract the number of 3-member committees and 5-member committees that contain Rocco as one common members.


For 3-member committees, the number of such committees is  {{{C[11]^2}}} = {{{(11*10)/2}}} = 11*5 = 55.


For 5-member committees, the number of such committees is  {{{C[11]^4}}} = {{{(11*10*9*8)/(1*2*3*4)}}} = 330.


Therefore, the <U>ANSWER</U>  to the problem's question is


    {{{C[12]^3}}}.{{{C[12]^5}}} - {{{C[11]^2}}}.{{{C[11]^4}}} = 174240  - 55*330 = 147015 ways.
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