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]
Rich Zwelling
GMAT Instructor, Veritas Prep
