# an n digit no is apositive no. with exactly n digit . Nine hundred distinct n digit no. r to be formed using only 3 digit 5,7.9. The smallest value of n is there r 4 balls of different clolur and 4 boxes colour same as those of balls the no of ways in which the ballon each  1 box could be plkaced such that any ball dosenot go to the box of its own colour

Sukhendra Reddy Rompally B.Tech Mining Machinery Engg, ISM Dhanbad
93 Points
14 years ago

Hi Anita Sharma!!!

1-the answer is 7,

explanation-u have the digits 5,7,9.and u have to form 900 distinct numbers.For any n digit number,the possible ways of filling each digit{that is units,tens,hundreds,etc} is 3

for a 1 digit number,total no. of ways is 3 power1=3{5,7,9}

for a 2 digit number,total no. of ways is 3 power 2+9{55,57,59,75,77,79,95,97,99}

for an n digit number,total no. of ways is 3 power n

now,we hav total no. of ways is 900,so we have to find a minimum n for which 3 power n is greater dan or equal to 900

3 power 6 is 729 and 3 power 7 is 2187,hence 7 is the answer

now the second ques

this is called dearrangements,and der is a forumla given in Dasgupta{which i dont remember now :(}

but,there is an alternative way,where u could choose cases where all 4 go to the ryt boxes,any 3,any2 and any 1 go into the right boxes and subtract the sum of all these cases from 4! which is the total no. of ways of arranging the balls{NOTE-THIS IS POSSIBLE ONLY ID THE NO. IS SMALL,LIKE 4 HERE,U CANT WASTE TYM TRYING TO SORT OUT CASES IF THE NO. OF BALLS/BOCES IS HIGH LIKE 7 OR 8,SO TRY TO CHECK THE FORMULA FOR DEARRANGEMENTS}

No. of ways off all balls goin into the ryt boxes is 1

No. of ways of any 3 balls going into the right boxes is zero as if 3 go into the right boxes,then the 4th will have to go into the right one

No. of ways of any 2 balls goin into the ryt box is 4c2 * 1! = 6*1=6{i.e; choosin 2 go into the ryt ones,and the total ways others go into the wrong ones,which is (2-1)!}

No. of ways of any 1 ball going into the ryt box is 4c1 * 2! =8

now,the sum of all these is 1+0+6+8=15

total ways of arranging is 4!=24

so,total ways of all balls going into the wrong boxes is 24-15 =9

Hope u understand the solutions

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