Question of the dated 05/07/10
Last Updated on Thursday, 08 July 2010 16:10 Written by TVM Thursday, 08 July 2010 05:32
Ques: A 6 digit number is formed by using digits 1.2.3.4.5 & 6. without repeating digits. What is the sum of all such no.s
a) 279999720 b) 299997290 c) 379999720
d) None of these
Soln: Simple Perm and comb first (to scare you). If a 6 digit number has to be formed using 6 digits (repetition not allowed) then this will be done in 6! Ways (=720, I used scientific calculator to find this out, so better don’t ask me how 720). Let’s take fedcba as a 6 digit number, where a, b, c, d, e, f only represents places and no significance w.r.t values.Now at place a, all the 6 digits 1 to6 can come.
Right!!Which will come more number of times and which will come less number of times? Yes you are right. All will come equal number of times. So how many times will they come.Since there are 720 such numbers and 6 digits for place a, I can say each digit therefore will come at place a, 720/6=120 times.So when we will start adding all the 720 numbers, can I say 1 will be added 120 times, similarly 2 will be added 120 times and so on.
So sum of the digits at place a will be 120 times 1 + 120 times 2 and so on.Or 120 x ( 1 + 2 + 3 + 4 + 5 + 6 ) = 120 x 21 = 2520.And in the same fashion can I say sum of the digits at place b will also be same and similarly at place c and so on.Understand this. 2520 at place a = 2520 x 12520 at place b = 2520 x 102520 at place c = 25 x 100And so onI hope this revised the concept of place value and face value which you learnt in class 1 and 2.
Mind it guys CAT is a test of class 1 to 10th Maths Mainly(Those who were scared on it, be happy. Those who are still not happy, wait. Shortcut is yet to come).So the sum of all those 720 numbers will be 2520 x 1 + 2520 x 10 + 2520 x 100 + 2520 x 1000 + 2520 x 10000 + 2520 x 100000 = 2520 x 111111 = 279999720 hence choice A.Shortcut(Only for those who like me hates perm and comb): No doubt there will be 720 numbers.
What’s the smallest of all? 123456 and Yes you are right for the first time. What’s the largest? 654321 and yes you are right for the 2nd time. What’s their average? (123456 + 654321) / 2 and yes you are right for the 3rd time. Now multiply this average by the number of numbers you have i.e. ( 777777) / 2 x 720 = 777777 * 360 = 279999720. Hence same option A. Hey the answer did not change. Surprising, isn’t it? We will think later how come answer didn’t change.
All the best
Regards
TVM
