How many different sums can be formed with the denominations Rs. 50, Rs. 100, Rs. 200, Rs. 500 and Rs. 2,000 taking at least three denominations at a time?

We have to choose at least three denominations out of five. So we can either choose 3 denominations out of 5 or choose 4 out of 5 or choose all 5.

Mathematically it will be written as –

Total number of ways = 5C3 + 5C4 + 5C5 = 10 + 5 + 1 = 16 ways.

Don’t worry about ‘different sums’, they will be ‘different’ as each time we are taking a different set of denominations.

Manually, these combinations will be –

(a) Three denomination combinations –

50+100+200; 50+100+500; 50+100+2000; 50+200+500; 50+200+2000; 50+500+2000; 100+200+500; 100+200+2000; 100+500+2000; 200+500+2000; TOTAL 10

(b) Four denomination combinations –

50+100+200+500; 50+100+200+2000; 50+100+500+2000; 50+200+500+2000; 100+200+500+2000; TOTAL 5

(c) Five denomination combinations –

50+100+200+500+2000; TOTAL 1

So total = 10 + 5 + 1 = 16.