Question

Suppose you have a currency named as MISO with denominations 1 MISO, 10 MISO, 50 MISO. In how many ways can you pay a bill of 107 MISOS.

Answer 1

Let the number of currency 1 Miso, 10 Misos and 50 Misos be x, y and z respectively.
then, x+10y+50z=107
Now the possible values of z could be 0, 1 and 2.

For z=0: x+10y=107
Number of integral pairs of values of x and y that satisfy the equation:
x+10y=107 will be 11.
These values of x and y in that order are:
(7,10);(17,9);(27,8)…(107,0)

For z=1: x+10y=57
Number of integral pairs of values of x and y that satisfy the equation:
x+10y=57 will be 6.
These values of x and y in that order are:
(7,5);(17,4);(27,3);(37,2);(47,1) and (57,0)

For z=2: x+10y=7
There is only one integer value of x and y that satisfies the equation:
x+10y=7 in that order is (7,0)

Therefore total number of ways in which you can pay a bill of 107 Misos:
=11+6+1= 18