Problem: There used to be unrest in the Galactic Senate. Several thousand solar systems had declared their intentions to leave the Republic. But fear not! Master Heidi was able to successfully select the Jedi Knights that have restored peace in the galaxy. However, she knows that evil never sleeps and a time may come when she will need to pick another group of Jedi Knights. She wants to be sure she has enough options to do so.

There are n Jedi Knights, each of them with a lightsaber of one of m colors. Given a number k, compute the number of differently colored collections of k lightsabers that some k Jedi Knights might have. Jedi Knights with lightsabers of the same color are indistinguishable (it's not the person, it's the lightsaber color that matters!), and their order does not matter; that is, we consider two collections of Jedi Knights to be different if and only if their vectors of counts of lightsabers of each color (like what you were given in the easy and the medium versions) are different. We count all subsets, not only contiguous subsegments of the input sequence. Output the answer modulo 1009.

Input

The first line of the input contains n (1 ≤ n ≤ 2·105), m (1 ≤ m ≤ n) and k (1 ≤ k ≤ n). The second line contains n integers in the range {1, 2, ..., m} representing colors of the lightsabers of subsequent Jedi Knights.

Output

Output one number: the number of differently colored collections of k lightsabers modulo 1009.

Example

Input

4 3 2
1 2 3 2


Output

4

Solution (in JAVA):

Solution:
import java.io.*;
import java.util.*;

public class E implements Runnable{
	public static void main (String[] args) {new Thread(null, new E(), "_cf", 1 << 28).start();}

	int MOD = 1009;
	int N, K;
	
	public void run() {
		DeoxysS fs = new DeoxysS();
//		try { fs = new DeoxysS("testdata.out"); } catch (Exception e) {}
		long start = System.nanoTime();
		PrintWriter out = new PrintWriter(System.out);
		System.err.println("Go!");
		
		N = fs.nextInt(); int m = fs.nextInt(); K = fs.nextInt();
		int[] freq = new int[m];
		for(int i = 0; i < N; i++)
			freq[fs.nextInt() - 1]++;

		int[][] poly = new int[m][];
		PriorityQueue<int[]> pq = new PriorityQueue<>((a, b) -> Integer.compare(a.length, b.length));
		for(int i = 0; i < m; i++) {
			poly[i] = new int[freq[i] + 1];
			Arrays.fill(poly[i], 1);
			pq.add(poly[i]);
		}
		
		FFT fft = new FFT();
		fft.maxk = Integer.numberOfTrailingZeros(Integer.highestOneBit(K) * 2);
		
		while(pq.size() > 1) {
			int[] a = pq.poll();
			int[] b = pq.poll();
			int[] mult = fft.multiply(a, b);
//			System.out.printf("%s * %s = %s\n", Arrays.toString(a), Arrays.toString(b), Arrays.toString(mult));
			if(mult.length > (1 << fft.maxk))
				mult = Arrays.copyOfRange(mult, 0, (1 << fft.maxk));
			pq.add(mult);
		}

		int[] ways = pq.poll();
		out.println(ways[K]);
		
//		System.err.println((System.nanoTime() - start) / 1000000);
		
		out.close();
	}
	
	int[] fftExpo(FFT fft, int[] it, int k) {
		if(k == 1) return it;
		int[] ret = fftExpo(fft, it, k / 2);
		ret = fft.multiply(ret, ret);
		if(ret.length > K + 1)
			ret = Arrays.copyOfRange(ret, 0, K + 1);
		if((k & 1) == 1)
			ret = fft.multiply(ret, it);
		if(ret.length > K + 1)
			ret = Arrays.copyOfRange(ret, 0, K + 1);
		return ret;
	}
	
	// SET maxk appropriately!!! ~log(maxn) //%
	class FFT {
	    int maxk = 21, maxn = (1 << maxk) + 1;
	    // Init: wR, wI, rR, rI, aR, aI   to   new double[maxn] !!!
	    //#
	    double[] wR = new double[maxn],
	            wI = new double[maxn],
	            rR = new double[maxn],
	            rI = new double[maxn],
	            aR  = new double[maxn],
	            aI  = new double[maxn]; //$
	    int n, k, lastk = -1, dp[] = new int[maxn];

	    void fft(boolean inv) {
	        if (lastk != k) {
	            lastk = k;  dp[0] = 0;
	            for (int i = 1, g = -1; i < n; i++) {
	                if ((i & (i - 1)) == 0) g++;
	                dp[i] = dp[i ^ (1 << g)] ^ (1 << (k - 1 - g));
	            }
	            wR[1] = 1;
	            wI[1] = 0;
	            for (int t = 0; t < k - 1; t++) {
	                double a = Math.PI / n * (1 << (k - 1 - t));
	                double curR = Math.cos(a), curI = Math.sin(a);
	                int p2 = (1 << t), p3 = p2 * 2;
	                for (int j = p2, k = j * 2; j < p3; j++, k += 2) {
	                    wR[k] = wR[j];
	                    wI[k] = wI[j];
	                    wR[k + 1] = wR[j] * curR - wI[j] * curI;
	                    wI[k + 1] = wR[j] * curI + wI[j] * curR;
	                }
	            }
	        }
	        for (int i = 0; i < n; i++) {
	            int d = dp[i];
	            if (i < d) {
	                double tmp = aR[i];
	                aR[i] = aR[d];
	                aR[d] = tmp;
	                tmp = aI[i];
	                aI[i] = aI[d];
	                aI[d] = tmp;
	            }
	        }
	        if (inv) for (int i = 0; i < n; i++) aI[i] = -aI[i];
	        for (int len = 1; len < n; len <<= 1) {
	            for (int i = 0; i < n; i += len) {
	                int wit = len;
	                for (int it = 0, j = i + len; it < len; it++, i++, j++) {
	                    double tmpR = aR[j] * wR[wit] - aI[j] * wI[wit];
	                    double tmpI = aR[j] * wI[wit] + aI[j] * wR[wit];
	                    wit++;
	                    aR[j] = aR[i] - tmpR;
	                    aI[j] = aI[i] - tmpI;
	                    aR[i] += tmpR;
	                    aI[i] += tmpI;
	                }
	            }
	        }
	    }

	    int[] multiply(int[] a, int[] b) {
	        int na = a.length, nb = b.length;
	        for (k = 0, n = 1; n < na + nb - 1; n <<= 1, k++) {}
	        for (int i = 0; i < n; ++i) {
	            aR[i] = i < na ? a[i] : 0;
	            aI[i] = i < nb ? b[i] : 0;
	        }
	        fft(false);
	        aR[n] = aR[0];
	        aI[n] = aI[0];
	        double q = -1.0 / n / 4.0;
	        for (int i = 0; i <= n - i; ++i) {
	            double tmpR = aR[i] * aR[i] - aI[i] * aI[i];
	            double tmpI = aR[i] * aI[i] * 2;
	            tmpR -= aR[n - i] * aR[n - i] - aI[n - i] * aI[n - i];
	            tmpI -= aR[n - i] * aI[n - i] * -2;
	            aR[i] = -tmpI * q;
	            aI[i] = tmpR * q;
	            aR[n - i] = aR[i];
	            aI[n - i] = -aI[i];
	        }
	        fft(true);
	        int[] ans = new int[n = na + nb - 1]; // ONLY MOD IF NEEDED
	        for (int i = 0; i < n; i++) ans[i] = (int) (Math.round(aR[i]) % MOD);
	        return ans;
	    }
	}

	class DeoxysS {
		public int BS = 1<<16;
		public char NC = (char)0;
		byte[] buf = new byte[BS];
		int bId = 0, size = 0;
		char c = NC;
		double num = 1;
		BufferedInputStream in;

		public DeoxysS() {
			in = new BufferedInputStream(System.in, BS);
		}

		public DeoxysS(String s) throws FileNotFoundException {
			in = new BufferedInputStream(new FileInputStream(new File(s)), BS);
		}

		public char nextChar(){
			while(bId==size) {
				try {
					size = in.read(buf);
				}catch(Exception e) {
					return NC;
				}                
				if(size==-1)return NC;
				bId=0;
			}
			return (char)buf[bId++];
		}

		public int nextInt() {
			return (int)nextLong();
		}

		public long nextLong() {
			num=1;
			boolean neg = false;
			if(c==NC)c=nextChar();
			for(;(c<'0' || c>'9'); c = nextChar()) {
				if(c=='-')neg=true;
			}
			long res = 0;
			for(; c>='0' && c <='9'; c=nextChar()) {
				res = (res<<3)+(res<<1)+c-'0';
				num*=10;
			}
			return neg?-res:res;
		}

		public double nextDouble() {
			double cur = nextLong();
			return c!='.' ? cur:cur+nextLong()/num;
		}

		public String next() {
			StringBuilder res = new StringBuilder();
			while(c<=32)c=nextChar();
			while(c>32) {
				res.append(c);
				c=nextChar();
			}
			return res.toString();
		}

		public String nextLine() {
			StringBuilder res = new StringBuilder();
			while(c<=32)c=nextChar();
			while(c!='\n') {
				res.append(c);
				c=nextChar();
			}
			return res.toString();
		}

		public boolean hasNext() {
			if(c>32)return true;
			while(true) {
				c=nextChar();
				if(c==NC)return false;
				else if(c>32)return true;
			}
		}
		
		public int[] nextIntArray(int n) {
			int[] res = new int[n];
			for(int i = 0; i < n; i++) res[i] = nextInt();
			return res;
		}
		
	}

}