Patchouli‘s Spell Cards
Time Limit: 7 Seconds
Memory Limit: 65536 KB
Patchouli Knowledge, the unmoving great library, is a magician who has settled down in the Scarlet Devil Mansion (紅魔館). Her specialty is elemental magic employing the seven elements fire, water, wood, metal, earth, sun, and moon. So she can cast different
spell cards like Water Sign "Princess Undine", Moon Sign "Silent Selene" and
Sun Sign "Royal Flare". In addition, she can combine the elements as well. So she can also cast high-level spell cards like
Metal & Water Sign "Mercury Poison" and Fire, Water, Wood, Metal & Earth Sign "Philosopher‘s Stones"
.
Assume that there are m different elements in total, each element has
n different phase. Patchouli can use many different elements in a single spell card, as long as these elements have the same phases. The level of a spell card is determined by the number of different elements used in it. When Patchouli is going to
have a fight, she will choose m different elements, each of which will have a random phase with the same probability. What‘s the probability that she can cast a spell card of which the level is no less than
l, namely a spell card using at least l different elements.
Input
There are multiple cases. Each case contains three integers 1 ≤ m,
n, l ≤ 100. Process to the end of file.
Output
For each case, output the probability as irreducible fraction. If it is impossible, output "mukyu~" instead.
Sample Input
7 6 5 7 7 7 7 8 9
Sample Output
187/15552 1/117649 mukyu~
题目意思：有m个元素，每个元素有n种相，现在从m个元素中，随机抽取每一种元素的一种相，每种相取到的概率相同，相同的相可以融合，融合的个数就为它的度，问你最大的度大于等于L的概率。
做法 ：
JAVA大数
dp[n][m] 表示的意思是 前n种相，用了m个元素，符合最大的度小于L的方案个数。
dp[n][m] = sum(dp[n-1][m-k]*count[M-(m-k)][k])
import java.util.*;
import java.math.*;
public class Main {
static int maxn = 110;
public static BigInteger[][] dp =new BigInteger[110][110];
public static BigInteger[][] count = new BigInteger[110][110];
public static void main(String[] args) {
// TODO Auto-generated method stub
for(int i = 0; i < maxn; i++){
count[i][0] = BigInteger.ONE;
count[i][i] = BigInteger.ONE;
for(int j = 1; j < i; j++){
count[i][j] = count[i-1][j].add(count[i-1][j-1]);
}
}
Scanner scan = new Scanner(System.in);
while(scan.hasNextInt()){
int m = scan.nextInt();
int n = scan.nextInt();
int l = scan.nextInt();
BigInteger nn = BigInteger.valueOf(n).subtract(BigInteger.ONE);
BigInteger fm = BigInteger.valueOf(n).pow(m),fz = BigInteger.ZERO;
if(l > m){
System.out.println("mukyu~");
}else{
for(int i = 0; i <= n; i++)
for(int j = 0; j <= m; j++)
dp[i][j] = BigInteger.ZERO;
dp[0][0] = BigInteger.ONE;
for(int i = 1; i <= n; i++){
for(int j = 0; j <= m; j++){
for(int k = 0; k <= j&& k <l; k++){
dp[i][j] = dp[i][j].add(dp[i-1][j-k].multiply(count[m-(j-k)][k]));
}
}
}
fz = fm.subtract(dp[n][m]);
BigInteger gcd = fz.gcd(fm);
fz = fz.divide(gcd);
fm = fm.divide(gcd);
System.out.println(fz+"/"+fm);
}
}
scan.close();
}
}
ZOJ3380- Patchouli's Spell Cards(概率DP+计数),布布扣,bubuko.com
ZOJ3380- Patchouli's Spell Cards(概率DP+计数)