In how many ways can 5 pounds be paid in shillings and sixpences, including zero solutions( positive integer solutions )?
1 pound = 20 shillings
1 pound = 40 sixpences
Let the number of shillings be x and the number of sixpences be y
According to question, we have
x20+y40=5⇒2x+y40=5⇒2x+y=200 .....(i)
We will approach the question by finding the number of positive integrals solution of (i) as the number of coins can only be integer.
x+y2=100
As x is an integer
⇒y2= integer
Let the integer be p
y2=p
⇒y=2p .....(ii)
Substituting y in (i), we get
2x+2p=200
⇒2x=200−2p
⇒x=100−p ......(iii)
From (ii) we can see that y<0 for integer p<0 and from (iii) x<0 for integer p>100, which is not possible
So p can take all integer values from [0,100]
So, the number of possible distributions is 101.