A seven digit number is in the form of abcdefg⁡(g$$,f,e, etc. are digits at units, tens, hundreds place etc.$$) where aefg . Then the number of such possible numbers is

Question

A seven digit number is in the form of abcdefg⁡(g$$,f,e, etc. are digits at units, tens,  hundreds place etc.$$) where ae>f>g . Then the number of such possible numbers is

Infinity Learn · Accepted Answer

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) .

A seven digit number is in the form of abcdefg⁡(g,f,e, etc. are digits at units, tens, hundreds place etc.) where ae>f>g . Then the number of such possible numbers is

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