How many ways can you make change for a dollar? This post points to two approaches to the problem, one computational and one analytic.
SICP gives a Scheme program to solve the problem:
(define (count-change amount) (cc amount 5)) (define (cc amount kinds-of-coins) (cond ((= amount 0) 1) ((or (< amount 0) (= kinds-of-coins 0)) 0) (else (+ (cc amount (- kinds-of-coins 1)) (cc (- amount (first-denomination kinds-of-coins)) kinds-of-coins))))) (define (first-denomination kinds-of-coins) (cond ((= kinds-of-coins 1) 1) ((= kinds-of-coins 2) 5) ((= kinds-of-coins 3) 10) ((= kinds-of-coins 4) 25) ((= kinds-of-coins 5) 50)))
Concrete Mathematics explains that the number of ways to make change for an amount of n cents is the coefficient of z^n in the power series for the following:
Later on the book gives a more explicit but complicated formula for the coefficients.
Both show that there are 292 ways to make change for a dollar.