Since the number of ways to choose 2-pound coins is fewer than the numbers of ways to do the other two coins, map out each possibility:

50 £2 coins - that already sums to £100 - 1 way

49 £2 coins - you could have 2 £1s; or you could have £1 and 2 50ps; or you could have0 £1 and 4 50ps - 3 ways

48 £2 coins - 4 £1s; 3 £1 and 2 50ps; 2 £1 and 4 50ps; 1 £1 and 6 50ps; 0 £1 and 8 50ps - 5 ways

47 £2 coins - 6 £1 coins ... ; 5 £1 coins ...; ... ... ... ; 0 £1 coins - 7 ways

You see the recurring pattern: You're progressing in consecutive odds. Continue the pattern:
...
...

1 £2 coin - 98 £1 coins; 97 £1 coins and 2 50ps; ... ... ... ; 0 £1 coin, - 99 ways

0 £2 coins - 100 £1 coins; 99 £1 coins and 2 50ps; ... ... ; 0 £1 coins and 200 50ps - 101 ways

Add up all the ways, and you get:

1+3+5+7+...+99+101

There are 51 numbers in that list. You can tell this because you're adding consecutive odd numbers. 1 = 2(0) + 1. 3 = 2(1)+1. And 101 = 2(50)+1. So you're adding all consecutive odds from 2(0)+1 to 2(50)+1. 51 terms.

You can group these into 50 pairs of numbers plus a 51 in the middle. Like so:

(101+1)+(99+3)+(97+5)+ ... + (53+49) + 51

That's 25 pairs of 102 plus the extra 51:

102 * 25 + 51 =

2601

You could also reach this number by identifying a pattern in sums of consecutive odds:

1 = 1
1+3 = 4
1+3+5 = 9
1+3+5+7 = 16

Notice that the sum of the first N consecutive positive odds is simply N^2.

Since we're adding the first 51 consecutive positive odds, our answer will simply be [spoiler]51^2 = 2601[/spoiler]

That's the correct answer.... Thanks alot for the reply and amazing explanation!

My pleasure

Hi Rich,I still dint get how did you get 51 pairs pls help me out with it!
Thanks
Raunak

It's actually not 51 pairs. It's 51 numbers total in the list:

1+3+5+7+...+99+101

1 is the 1st positive odd number
3 is the 2nd positive odd number
5 is the 3rd positive odd number
...
2n-1 is the nth positive odd number
...
101 is the 51st positive odd number (since 2n-1 = 2(51) - 1 = 101)

Since there are 51 numbers in the list, there will be 25 pairs of numbers plus one left over.

Make sense?

yes thanks a ton Rich!