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 Ruby (2.3.3) (judge verdict: Accepted)

N = gets.chomp.chars.map(&:to_i)
K = gets.to_i

DIGITS = N.size
dp = Array.new(DIGITS + 1) { Array.new(2) { Array.new(K + 2, 0) } }
dp[0][0][0] = 1
DIGITS.times do |i|
  d = N[i]
  # same
  if d == 0
    (0..K).each do |k|
      dp[i + 1][0][k] = dp[i][0][k]
    end
  else
    (0..K).each do |k|
      c = 0
      (0..i).each do |ii|
        c += 1 if N[ii] != 0
      end
      dp[i + 1][0][c] = 1
    end
  end
  # lower
  (0..9).each do |n|
    (0..K).each do |k|
      if n == 0
        dp[i + 1][1][k] += dp[i][1][k]
      else
        dp[i + 1][1][k + 1] += dp[i][1][k]
      end
    end
  end
  # same -> lower
  (0..d).each do |n|
    next if n == d
    (0..K).each do |k|
      if n == 0
        dp[i + 1][1][k] += dp[i][0][k]
      else
        dp[i + 1][1][k + 1] += dp[i][0][k]
      end
    end
  end
end

ans = dp[DIGITS][0][K] + dp[DIGITS][1][K]
puts ans