Problem:
Score : 600 points

Problem Statement

AtCoDeer the deer has N cards with positive integers written on them. The number on the i-th card (1≤i≤N) is a_i.
Because he loves big numbers, he calls a subset of the cards good when the sum of the numbers written on the cards in the subset, is K or greater.

Then, for each card i, he judges whether it is unnecessary or not, as follows:

If, for any good subset of the cards containing card i, the set that can be obtained by eliminating card i from the subset is also good, card i is unnecessary.

Otherwise, card i is NOT unnecessary.

Find the number of the unnecessary cards. Here, he judges each card independently, and he does not throw away cards that turn out to be unnecessary.

Constraints

All input values are integers.

1≤N≤5000

1≤K≤5000

1≤a_i≤10^9 (1≤i≤N)

Partial Score

300 points will be awarded for passing the test set satisfying N,K≤400.

Input

The input is given from Standard Input in the following format:

N K
a_1 a_2 ... a_N

Output

Print the number of the unnecessary cards.

Sample Input 1

3 6
1 4 3

Sample Output 1

1

There are two good sets: {2,3} and {1,2,3}.

Card 1 is only contained in {1,2,3}, and this set without card 1, {2,3}, is also good. Thus, card 1 is unnecessary.

For card 2, a good set {2,3} without card 2, {3}, is not good. Thus, card 2 is NOT unnecessary.

Neither is card 3 for a similar reason, hence the answer is 1.

Sample Input 2

5 400
3 1 4 1 5

Sample Output 2

5

In this case, there is no good set. Therefore, all the cards are unnecessary.

Sample Input 3

6 20
10 4 3 10 25 2

Sample Output 3

3

Solution in Ruby (2.3.3) (judge verdict: Wrong Answer)

def partial_sum_dp(items, sum)
    n_items = items.length
    dp = Array.new(n_items){Hash.new}
    dp[0][0] = true
    dp[0][items[0]] = true if items[0] <= sum
    (1...n_items).each do |i|
        (0..sum).each do |j|
            dp[i][j] = true if dp[i - 1][j] ||  dp[i - 1][j - items[i]]
        end
    end
    return dp
end



N, K = gets.split.map(&:to_i)
as = gets.split.map(&:to_i).sort

if as.length == 1
    puts as[0] >= K ? 0 : 1
    exit
end

dp_normal = partial_sum_dp(as, K)
asr = as.reverse
dp_reverse = partial_sum_dp(asr, K).reverse

need_min = N.times.to_a.bsearch do |i|
    a = as[i]
    sum1_cand = i > 0 ? dp_normal[i - 1] : {0 => true}
    sum2_cand = i < N - 1 ? dp_reverse[i + 1] : {0 => true}
    lower = [K - a, 0].max
    upper = K - 1

    flag = false

    sum2_accum = Array.new(upper + 1, 0)
    (0..upper).each do |i|
        sum2_accum[i + 1] = sum2_accum[i] + (sum2_cand[i + 1] ? 1 : 0)
    end

    sum1_cand.each do |sum1, _|
        break if flag
        sum2_lower = [lower - sum1, 0].max
        sum2_upper = [upper - sum1, 0].max
        if sum2_accum[sum2_lower] != sum2_accum[sum2_upper]
            flag = true
            break
        end
    end

    flag
end

puts need_min || N