How many numbers are between 1 and 99999 which have the sum of their digits equal to 9? I need the method.
2 Answers
Explanation:
Explanation:
Suppose you have
Writing
-
#""^0 D_k = 1# -
#""^1 D_k = k# -
#""^n D_1 = 1# -
#""^n D_2 = ""^n D_1 + ""^(n-1) D_1 + ... +""^0 D_1 = n+1# -
#""^n D_3 = ""^n D_2 + ""^(n-1) D_2 + ... + ""^0 D_2# #= (n+1)+((n-1)+1) + ... + (1+1) + (0+1) = 1/2 (n+1)(n+2)# -
#""^n D_4 = ""^n D_3 + ""^(n-1) D_3 + ... + ""^0 D_3# #= 1/2(n+1)(n+2) + 1/2((n-1)+1)((n-1)+2) + ... + 1/2(0+1)(0+2)#
#= 1/6(n+1)(n+2)(n+3)#
#""^n D_5 = ""^n D_4 + ""^(n-1) D_4 + ... + ""^0 D_4# #= 1/6(n+1)(n+2)(n+3)+1/6((n-1)+1)((n-1)+2)((n-1)+3)+...+1/6(0+1)(0+2)(0+3)#
#= 1/24(n+1)(n+2)(n+3)(n+4)#
So:
#""^9 D_5 = 1/24(9+1)(9+2)(9+3)(9+4) = 715#