The sum of all the numbers that can be formed by writing all the digits $$3$$, $$2$$, $$3$$, $$4$$ only once is

Question

Infinity Learn · Accepted Answer

The number of numbers having $$2$$ in unit’s place $$=3$$!$$2$$!(because$$ the other three places are to be filled by $$3$$, $$3$$, $$4$$ of which two are $$identical).The number of numbers having $$3$$ in unit’s place $$=$$ $$3$$!, (because$$ the other three places are to be filled by $$2$$, $$3$$, $$4).The number of numbers having $$4$$ in unit’s place $$=3$$!$$2$$!, (because$$ the other three places are to be filled by $$3$$, $$2$$, $$3$$ of which two are $$identical)∴ Sum of the digits in unit’s $$place=3$$!$$2$$!×$$2+3$$!×$$3+3$$!$$2$$!×$$4=6+18+12=36$$.Similarly, the sum of the digits in other places is $$36$$ each.∴ Required sum $$=$$ $$36$$ × $$1000$$ $$+$$ $$36$$ × $$100$$ $$+$$ $$36$$ × $$10$$ $$+$$ $$36$$ × $$1$$                          $$=$$ $$36$$ (1000$$ $$+$$ $$100$$ $$+$$ $$10$$ $$+1) $$=$$ $$36$$ × $$1111$$ $$=$$ $$39996$$.

The sum of all the numbers that can be formed by writing all the digits 3, 2, 3, 4 only once is

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material