$\text{ A seven digit number is in the form of }abc\mathrm{defg}(g,f,e,\text{ etc. are digits at units, tens, }$

$\text{ hundreds place etc.) where }a<b<c<d>e>f>g\text{ . Then the number of such possible }$numbers is

detailed solution

Correct option is C

The minimum value of  d will be 4 i) If d=4 , then the number of seven digit numbers possible is 3C3⋅4C3=4 [as a,b,c can be chosen from 1,2 or 3 and similarly e,f,g can be chosen from 0,1,2 or 3 ]  ii) If d=5 , then the number of seven digit numbers possible is 4C3⋅5C3=40 [as a,b,c can be chosen from 1,2,3 or 4 and similarly e,f,g can be chosen from 0,1,2,3 or4 ] iii) If d=6 , then the number of seven digit numbers possible is 5C3⋅6C3=200 [as a,b,c can be chosen from 1,2,3,4 or 5 and similarly e,f,g can be chosen from 0,1,2,3,4 or 5] iv) If d=7 , then the number of seven digit numbers possible is 6C3⋅7C3=700 [as a,b,c can be chosen from 1,2,3,4,5 or 6 and similarly e,f,g can be chosen from 0,1,2,3,4,5,6 or 7]  v) If d=8 , then the number of seven digit numbers possible is 7C3⋅8C3=1960 [as a,b,c can be chosen from 1,2,3,4,5,6 or 7 and similarly e,f,g can be chosen from 0,1,2,3,4,5,6 or 7]  vi) If d=9 , then the number of seven digit numbers possible is 8C3⋅9C3=4704 [as a,b,c can be chosen from 1,2,3,4,5,6,7 or 8 and similarly e,f,g can be chosen from 0,1,2,3,4,5,6,7 or 8] Total number of numbers is 4+40+200+700+1960+4704=7608  Therefore, the correct answer is (3) .