Problem:
Score : 500 points

Problem Statement

Find the number of integers between 1 and N (inclusive) that contains exactly K non-zero digits when written in base ten.

Constraints

1 \leq N < 10^{100}

1 \leq K \leq 3

Input

Input is given from Standard Input in the following format:

N
K

Output

Print the count.

Sample Input 1

100
1

Sample Output 1

19

The following 19 integers satisfy the condition:

1,2,3,4,5,6,7,8,9,10,20,30,40,50,60,70,80,90,100

Sample Input 2

25
2

Sample Output 2

14

The following 14 integers satisfy the condition:

11,12,13,14,15,16,17,18,19,21,22,23,24,25

Sample Input 3

314159
2

Sample Output 3

937

Sample Input 4

9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
3

Sample Output 4

117879300

Solution in Python (3.4.3) (judge verdict: Accepted)

N=input()
K=int(input())
S="0"*(100-len(N))+N
DP=[[[0]*2 for _ in range(K+2)] for _ in range(101)]

DP[0][0][0] = 1
for i in range(100):
  for j in range(K+1):
    #SがN以下かわからないまま、「0」の文字をなぞるとき、0でない数字jはそのまま
    if S[i]=='0':
      DP[i+1][j][0]+=DP[i][j][0]
      
    #SがN以下かわからないまま、「0」以外の文字をなぞるとき、0でない数字jも+1
    else:
      DP[i+1][j+1][0]+=DP[i][j][0]
    
    #SがN以下が確定している　
    DP[i+1][j][1]+=sum(DP[i][j])
    if int(S[i])==0:
      #あとは数字が同じとはいえ000…と埋めるしかないパターン
      DP[i+1][j][1]-=DP[i][j][0]
      
    #SがN以下が確定しているものは、1~9をくっつけたものと、同じ桁だけどより低い数字が来たとき
    DP[i+1][j+1][1] += DP[i][j][1]*9+DP[i][j][0]*(max(0, int(S[i])-1))
    
print(DP[-1][K][0]+DP[-1][K][1])