I doubt it could be solved in under 3minutes!.. Please let me know if there is a shorter method

for 100s, we have following palindromes... 101 and 111,121, 131.... 191. We can split this as 100+11, 100+21, 100+31.. sum of this is equal to 100*9 + 11+21+...91

Similarly for 2s, we have 202, 212, 222, 232, 242... 292.. We can again split this as 200+11+1, 200+21+2, 200+11+3... This will result in 200 + 11, 200 +21... 200 +91, + 9. Sum of this is 200*9 + 11+21+31...+91 + 9

For 300s, this will result in 300+11, 300+21, 300+21 + 2*9..

Going for other numbers, we will find a similar pattern. So we will have 4 APs altogether...

101+202+303+...+909 ; 900+1800+2700...+8100; 11+21+31+...+91; and 0+9+18+..+72. Note that 11+21+31+..+91 series is repeated 9 times so we need to multiply sum of this series by 9.

Sum of AP = Middle term * no. of terms

Hence,

Sum of
AP 1 = 505*9; AP 2 = 4500*9; AP 3 = 51*9*9; AP 4 = 36*9

= 9 * (505+4500+459+35)
= 9* (5500)
= 495000
(C) IMO

There will be 9 such series 1a1, 2a2 …. 9a9

For the first one - there will be 10 numbers and the sum can be written as 100*(1+1…. 10 times)+ 10*(0+1+2…9) + (1+1….. 10 times)

For the others, the form will be similar

100x (2+2+2… 10 times) + 10x (0+1+2….9) + (2+2… 10 times)

So the sum can be written as 100*10*(1+2+3….9) + 10*9*(1+2+3…..9) + 10*(1+2…..9)

= 1000*45 + 90*45 + 10*45 = 45000 + 4050 + 450 = 49500 …. (C)

Posted from my mobile device

[quote="Bunuel"]A palindrome is a number that reads the same backward and forward, such as 545. What is the sum of all three-digit palindromes?

A. 49320
B. 49410
C. 49500
D. 49590
E. 49680

101+111+121+...............+191
202+212+222+...............+292
303+313+323+...............+393
....+...+...+.....................+...
....+...+...+......................+...
909+919+929+................+999
(Each Column is in AP with c.d. =101 & has 9 terms, while Each row is in AP with c.d. 10 & has 10 terms)
......................................................

(505*9)+(515*9)+(525*9)............(595*9)
=9(505+515+...595)
=9*(545+555)*10/2
=49500
Answer C.

**I took way long time to arrive at this solution **