An engagement team consists of a project manager, team leader, and four consultants. There are 2 candidates for the position of project manager, 3 candidates for the position of team leader, and 7 candidates for the 4 consultant slots. If 2 out of 7 consultants refuse to be on the same team, how many different teams are possible?  Options:
A) 25
B) 35
C) 150
D) 210
E) 300
I get C. If I see this on the test, id prolly get to 210 and then guess A B or C.   OK so for the first position we have only 2 possibilities ( proj manager) 3 for the team leader so and 7!/3!4! for the consultants   2*3*35 --&gt; 210   Now I dunno how to figure out the constraints quickly but I eventually figured it out. Obvs. we want to figure out the total ways in which the two ARE on the same team.   I did it by AB are the two and XYZFN are the rest   ABXO (O stands for the other 4) So the first is ABXY, Z,F,N 4 possible choices   The next is   ABYO (but notice we already had YX so there are only 3 possible choices)   ABZO   ABFO   ABNO No possible here we used them all up   So  4+3+2+1= 
<gadget id="calculator"> 4+3+2+1</gadget>
<output>10</output>
10   SO now its   2*3*10= 
<gadget id="calculator">  2*3*10</gadget>
<output>60</output>
60   So 210-60= 
<gadget id="calculator">210-60</gadget>
<output>150</output>
150   ANS:150
<result>C</result>
An engagement team consists of a project manager, team leader, and four consultants. There are 2 candidates for the position of project manager, 3 candidates for the position of team leader, and 7 candidates for the 4 consultant slots. If 2 out of 7 consultants refuse to be on the same team, how many different teams are possible?