Problem:


Score : 1200 points

Problem Statement

In Republic of AtCoder, Snuke Chameleons (Family: Chamaeleonidae, Genus: Bartaberia) are very popular pets.
Ringo keeps N Snuke Chameleons in a cage.

A Snuke Chameleon that has not eaten anything is blue. It changes its color according to the following rules:

A Snuke Chameleon that is blue will change its color to red when the number of red balls it has eaten becomes strictly larger than the number of blue balls it has eaten.

A Snuke Chameleon that is red will change its color to blue when the number of blue balls it has eaten becomes strictly larger than the number of red balls it has eaten.

Initially, every Snuke Chameleon had not eaten anything. Ringo fed them by repeating the following process K times:

Grab either a red ball or a blue ball.

Throw that ball into the cage. Then, one of the chameleons eats it.

After Ringo threw in K balls, all the chameleons were red. We are interested in the possible ways Ringo could have thrown in K balls. How many such ways are there? Find the count modulo 998244353. Here, two ways to throw in balls are considered different when there exists i such that the color of the ball that are thrown in the i-th throw is different.

Constraints

1 \leq N,K \leq 5 \times 10^5

N and K are integers.

Input

Input is given from Standard Input in the following format:

N K

Output

Print the possible ways Ringo could have thrown in K balls, modulo 998244353.

Sample Input 1

2 4

Sample Output 1

7

We will use R to represent a red ball, and B to represent a blue ball. There are seven ways to throw in balls that satisfy the condition: BRRR, RBRB, RBRR, RRBB, RRBR, RRRB and RRRR.

Sample Input 2

3 7

Sample Output 2

57

Sample Input 3

8 3

Sample Output 3

0

Sample Input 4

8 10

Sample Output 4

46

Sample Input 5

123456 234567

Sample Output 5

857617983



Sample input:
2 4


Sample output:
7

Submission status: Wrong Answer

Solution code:
def combs_mod(n,k,mod):
  #nC0からnCkまで
  inv = [1]*(k+1)
  for i in range(1,k+1):
    inv[i] = pow(i,mod-2,mod)
  ans = [1]*(k+1)
  for i in range(1,k+1):
    ans[i] = ans[i-1]*(n+1-i)*inv[i]%mod
  return ans
def solve():
  mod = 998244353
  N, K = map(int, input().split())
  ans = 0
  if K<N:
    return ans
  com = combs_mod(K,K,mod)
  com2 = combs_mod(K-1,K-1,mod)
  for r in range(K+1):
    b = K-r
    if r<b or r<N:
      continue
    elif r==b:
      ans += com2[r]
      if N>=2:
        ans -= com2[N-2]
    elif r-b<N:
      ans += com[r] - com[max(b,N)-1+b-r]
    else:
      ans += com[r]
    ans %= mod
  return ans
print(solve())

Submission status: Wrong Answer

Solution code:
m=998244353
def c(n,r):return (g[n]*h[r]*h[n-r])%m
g=[1,1]
h=g[:]
v=[0,1]
for i in range(2,999999):
  g.append((g[-1]*i)%m)
  v.append((-v[m%i]*(m//i))%m)
  h.append((h[-1]*v[i])%m)
N,K=map(int,input().split())
P=0
for k in range(N-1,K):P+=c(K-1,k)
print(P)

Submission status: Wrong Answer

Solution code:
def combs_mod(n,k,mod):
  #nC0からnCkまで
  inv = [1]*(k+1)
  for i in range(1,k+1):
    inv[i] = pow(i,mod-2,mod)
  ans = [1]*(k+1)
  for i in range(1,k+1):
    ans[i] = ans[i-1]*(n+1-i)*inv[i]%mod
  return ans
def solve():
  mod = 998244353
  N, K = map(int, input().split())
  ans = 0
  if K<N:
    return ans
  com = combs_mod(K,K,mod)
  for r in range(K+1):
    b = K-r
    if r<b or r<N:
      continue
    ans += com[r]
    if r-b<N:
      ans -= com[max(b,N)-1+b-r]
    ans %= mod
  return ans
print(solve())

Submission status: Wrong Answer

Solution code:
mod = 998244353

max_kai = 10**6
kai = [1]
for i in range(1, max_kai):
    kai.append(kai[-1] * i % mod)
kkai = [0] * max_kai
kkai[-1] = pow(kai[-1], mod - 2, mod)
for i in reversed(range(1, max_kai)):
    kkai[i - 1] = kkai[i] * i % mod


def comb(a, b):
    if a < 0 or b < 0:
        return 0
    elif a < b:
        return 0
    c = 1
    c *= kai[a]
    c *= kkai[b]
    c *= kkai[a - b]
    return c % mod


n, k = map(int, input().split())

ans = 0
for r in range(k+1):
    b = k - r
    if r < b:
        continue
    ans += comb(k, r)
    if r - b < n:
        ans -= comb(k, 2*r - n + 1)

print(ans % mod)

Submission status: Wrong Answer

Solution code:
def solve(n, k):
    MOD = 998244353

    if n > k:
        return 0
    if n == 1:
        return 1 << (k - 1)

    invs = [1] * (k + 1)
    pf, kf = 1, 1
    for m in range(2, k + 1):
        pf = kf
        kf *= m
        kf %= MOD
        invs[m] = pow(kf, MOD - 2, MOD)

    ans = 0
    if k & 1 == 0:
        r = k >> 1
        s = k - n + 1
        ans = pf * (invs[r] * invs[r - 1] - invs[s] * invs[k - s - 1]) % MOD

    for r in range(k // 2 + 1, k + 1):
        if r * 2 >= n + k:
            ans += kf * invs[r] * invs[k - r]
        else:
            s = r * 2 - n + 1
            ans += kf * (invs[r] * invs[k - r] - invs[s] * invs[k - s])
        ans %= MOD
    return ans


print(solve(*map(int, input().split())))

Submission status: Wrong Answer

Solution code:
def solve(n, k):
    MOD = 998244353

    if n > k:
        return 0

    invs = [1] * (k + 1)
    pf, kf = 1, 1
    for m in range(2, k + 1):
        pf = kf
        kf *= m
        kf %= MOD
        invs[m] = pow(kf, MOD - 2, MOD)

    ans = 0
    if k & 1 == 0:
        if n == 1:
            ans = 1
        else:
            r = k >> 1
            t = r - n + 1
            ans = pf * (invs[r] * invs[r - 1] - invs[r + t] * invs[k - r - t - 1]) % MOD

    for r in range(k // 2 + 1, k + 1):
        if r * 2 >= n + k:
            ans += kf * invs[r] * invs[k - r]
        else:
            t = r - n + 1
            ans += kf * (invs[r] * invs[k - r] - invs[r + t] * invs[k - r - t])
        ans %= MOD
    return ans


print(solve(*map(int, input().split())))

Submission status: Wrong Answer

Solution code:
def solve(n, k):
    MOD = 998244353

    if n > k:
        return 0

    facts = [1] * (k + 1)
    invs = [1] * (k + 1)
    kf = 1
    for m in range(2, k + 1):
        kf *= m
        kf %= MOD
        facts[m] = kf
        invs[m] = pow(kf, MOD - 2, MOD)

    ans = 0
    if k & 1 == 0:
        r = k >> 1
        t = r - n + 1
        ans = facts[k - 1] * (invs[r] * invs[r - 1] - invs[r + t] * invs[r - t - 1]) % MOD

    for r in range(k // 2 + 1, k + 1):
        if r * 2 >= n + k:
            ans += facts[k] * invs[r] * invs[k - r]
        else:
            t = r - n + 1
            ans += facts[k] * (invs[r] * invs[k - r] - invs[r + t] * invs[k - r - t])
        ans %= MOD
    return ans


print(solve(*map(int, input().split())))

Submission status: Time Limit Exceeded

Solution code:
mod = 998244353
def comb(n, r):
    if n < r:return 0
    if n < 0 or r < 0:return 0
    return fa[n] * fi[r] % mod * fi[n - r] % mod
n, k = map(int, input().split())
fa = [1] * (k + 1)
fi = [1] * (k + 1)
for i in range(1, k + 1):
    fa[i] = fa[i - 1] * i % mod
    fi[i] = pow(fa[i], mod - 2, mod)
ans = 0
for i in range(k - n + 1):
    ans += comb(k - 1, n + i - 1)
    ans %= mod
print(ans)

Submission status: Time Limit Exceeded

Solution code:
n,k=map(int,input().split())
p=998244353
r=range
f=[1]
for i in r(k):f+=[-~i*f[i]%p]
a=1<<~-k
for i in r(min(n-1,k)):a-=f[k-1]*pow((f[i]*f[k-1-i])%p,-1,p)
print(a%p)

Submission status: Time Limit Exceeded

Solution code:
n,k=map(int,input().split())
p=998244353
r=range
f=[1]
for i in r(k):f+=[(-~i*f[i])%p]
a=1<<~-k
for i in r(min(n-1,k)):a-=f[k-1]*pow(f[i]*f[k-1-i],-1,p)
print(a%p)

Submission status: Time Limit Exceeded

Solution code:
n,k=map(int,input().split())
p=998244353
r=range
f=[1]
for i in r(k):f+=[-~i*f[i]%p]
a=1<<~-k
for i in r(min(n-1,k)):a-=f[k-1]*pow(f[i]*f[k-1-i],-1,p)
print(a%p)

Submission status: Time Limit Exceeded

Solution code:
def combs_mod(n,k,mod):
  #nC0からnCkまで
  inv = [1]*(k+1)
  for i in range(1,k+1):
    inv[i] = pow(i,mod-2,mod)
  ans = [1]*(k+1)
  for i in range(1,k+1):
    ans[i] = ans[i-1]*(n+1-i)*inv[i]%mod
  return ans
def solve():
  mod = 998244353
  N, K = map(int, input().split())
  ans = 0
  if K<N:
    return ans
  com = combs_mod(K,K,mod)
  com2 = combs_mod(K-1,K-1,mod)
  for r in range(K+1):
    b = K-r
    dif = r-b
    if dif<0 or r<N:
      continue
    elif dif==0:
      ans += com2[r]
      if N>=2:
        ans -= com2[N-2]
    elif dif<N:
      ans += com[r] - com[N-1-dif]
    else:
      ans += com[r]
    ans %= mod
  return ans
print(solve())

Submission status: Time Limit Exceeded

Solution code:
n,k = map(int,input().split())
table = [[0 for i in range(k+1)]for e in range(n+1)]
table[1][1] = 1 #n,kの順
for i in range(2,k+1):
	table[1][i] = (table[1][i-1]*2)%998244353
for i in range(2,k+1):
	for e in range(2,n+1):
		table[e][i] = (table[e][i-1] + table[e-1][i-1])%998244353
print(table[n][k])

Submission status: Accepted

Solution code:
n,k=map(int,input().split())
p=998244353
r=range
f=[1]
for i in r(k):f+=[-~i*f[i]%p]
print(sum(f[-2]*pow(f[i]*f[-2-i],-1,p)for i in r(n-1,k))%p)

Submission status: Accepted

Solution code:
n,k=map(int,input().split());p=998244353;r=range;f=[1]
for i in r(k):f+=[-~i*f[i]%p]
print(sum(f[-2]*pow(f[i]*f[-2-i],-1,p)for i in r(n-1,k))%p)

Submission status: Accepted

Solution code:
#メモ代わりの写経
n,k=map(int,input().split())
p=998244353
f=[1]
#階乗のリストを先に作っておく
for i in range(k):
    #f+=[-~i*f[i]%p]
    f+=[(i+1)*f[i]%p]
    
#combinationsの総和    
print(sum(f[-2]*pow(f[i]*f[-2-i],-1,p)for i in range(n-1,k))%p)

Submission status: Accepted

Solution code:
n,k=map(int,input().split())
p=998244353
f=[1]
for i in range(k):f+=[-~i*f[i]%p]
print(sum(f[-2]*pow(f[i]*f[-2-i],-1,p)for i in range(n-1,k))%p)

Submission status: Accepted

Solution code:
n,k=map(int,input().split());p=998244353
r=range
f=[1]
for i in r(k):f+=[-~i*f[i]%p]
print(sum(f[-2]*pow(f[i]*f[-2-i],-1,p)for i in r(n-1,k))%p)

Submission status: Accepted

Solution code:
n,k=map(int,input().split())
p=998244353
r=range
f=[1]
for i in r(k):f+=[-~i*f[i]%p]
print(sum(f[k-1]*pow(f[i]*f[k-1-i],-1,p)for i in r(n-1,k))%p)

Submission status: Accepted

Solution code:
n,k=map(int,input().split())
p=998244353
r=range
f=[1]
for i in r(k):f+=[-~i*f[i]%p]
a=0
for i in r(n-1,k):a+=f[k-1]*pow(f[i]*f[k-1-i],-1,p)
print(a%p)

Submission status: Accepted

Solution code:
def combs_mod(n,k,mod):
  #nC0からnCkまで
  inv = [1]*(k+1)
  for i in range(1,k+1):
    inv[i] = pow(i,mod-2,mod)
  ans = [1]*(k+1)
  for i in range(1,k+1):
    ans[i] = ans[i-1]*(n+1-i)*inv[i]%mod
  return ans
def solve():
  mod = 998244353
  N, K = map(int, input().split())
  ans = 0
  if K<N:
    return ans
  com = combs_mod(K,K,mod)
  for r in range(K+1):
    b = K-r
    dif = r-b
    if dif<0 or r<N:
      continue
    elif dif==0:
      com2 = combs_mod(K-1,K-1,mod)
      ans += com2[r]
      if N>=2:
        ans -= com2[N-2]
    elif dif<N:
      ans += com[r] - com[N-1-dif]
    else:
      ans += com[r]
    ans %= mod
  return ans
print(solve())

Submission status: Accepted

Solution code:
def combs_mod(n,k,mod):
  #nC0からnCkまで
  inv = [1]*(k+1)
  for i in range(1,k+1):
    inv[i] = pow(i,-1,mod)
  ans = [1]*(k+1)
  for i in range(1,k+1):
    ans[i] = ans[i-1]*(n+1-i)*inv[i]%mod
  return ans
def solve():
  mod = 998244353
  N, K = map(int, input().split())
  ans = 0
  if K<N:
    return ans
  com = combs_mod(K,K,mod)
  com2 = combs_mod(K-1,K-1,mod)
  for r in range(K+1):
    b = K-r
    dif = r-b
    if dif<0 or r<N:
      continue
    elif dif==0:
      ans += com2[r]
      if N>=2:
        ans -= com2[N-2]
    elif dif<N:
      ans += com[r] - com[N-1-dif]
    else:
      ans += com[r]
    ans %= mod
  return ans
print(solve())

Submission status: Accepted

Solution code:
m=998244353
def c(n,r):return (g[n]*h[r]*h[n-r])%m
g=[1,1]
h=g[:]
v=[0,1]
for i in range(2,999999):
  g.append(g[-1]*i%m)
  v.append(m//i*-v[m%i]%m)
  h.append(h[-1]*v[i]%m)
N,K=map(int,input().split())
P=0
for k in range(N-1,K):P+=c(K-1,k)
print(P%m)

Submission status: Accepted

Solution code:
m=998244353
def c(n,r):return (g[n]*h[r]*h[n-r])%m
g=[1,1]
h=g[:]
v=[0,1]
for i in range(2,999999):
  g.append((g[-1]*i)%m)
  v.append((-v[m%i]*(m//i))%m)
  h.append((h[-1]*v[i])%m)
N,K=map(int,input().split())
P=0
for k in range(N-1,K):P+=c(K-1,k)
print(P%m)

Submission status: Accepted

Solution code:
mod=998244353

def cmb(n,r):
  if r<0 or r>n:
    return 0
  return (g1[n]*g2[r]*g2[n-r])%mod

g1=[1,1]
g2=[1,1]
inv=[0,1]
for i in range(2,1000003):
  g1.append((g1[-1]*i)%mod)
  inv.append((-inv[mod%i]*(mod//i))%mod)
  g2.append((g2[-1]*inv[-1])%mod)

N,K=map(int,input().split())
P=0
for k in range(N-1,K):
  P=(P+cmb(K-1,k))%mod
print(P)

Submission status: Accepted

Solution code:
def solve(n, k):
    MOD = 998244353

    if n > k:
        return 0
    if n == 1:
        return pow(2, k - 1, MOD)

    pf, kf = 1, 1
    for m in range(2, k + 1):
        pf = kf
        kf *= m
        kf %= MOD
    inv = pow(kf, MOD - 2, MOD)
    invs = [1] * (k + 1)
    invs[k] = inv
    for m in range(k, 1, -1):
        inv *= m
        inv %= MOD
        invs[m - 1] = inv

    ans = 0
    if k & 1 == 0:
        r = k >> 1
        s = k - n + 1
        ans = pf * (invs[r] * invs[r - 1] - invs[s] * invs[k - s - 1]) % MOD

    for r in range(k // 2 + 1, k + 1):
        if r * 2 >= n + k:
            ans += kf * invs[r] * invs[k - r]
        else:
            s = r * 2 - n + 1
            ans += kf * (invs[r] * invs[k - r] - invs[s] * invs[k - s])
        ans %= MOD

    return ans


print(solve(*map(int, input().split())))

Submission status: Accepted

Solution code:
def solve(n, k):
    MOD = 998244353

    if n > k:
        return 0
    if n == 1:
        return pow(2, k - 1, MOD)

    invs = [1] * (k + 1)
    pf, kf = 1, 1
    for m in range(2, k + 1):
        pf = kf
        kf *= m
        kf %= MOD
        invs[m] = pow(kf, MOD - 2, MOD)

    ans = 0
    if k & 1 == 0:
        r = k >> 1
        s = k - n + 1
        ans = pf * (invs[r] * invs[r - 1] - invs[s] * invs[k - s - 1]) % MOD

    for r in range(k // 2 + 1, k + 1):
        if r * 2 >= n + k:
            ans += kf * invs[r] * invs[k - r]
        else:
            s = r * 2 - n + 1
            ans += kf * (invs[r] * invs[k - r] - invs[s] * invs[k - s])
        ans %= MOD
    return ans


print(solve(*map(int, input().split())))

Submission status: Runtime Error

Solution code:
n,k,p=map(int,input().split()),998244353
r=range
f=[1]
for i in r(k):f+=[-~i*f[i]]
print(sum(f[-1]*pow(f[i]*f[-1-i],-1,p)for i in r(n-1,k))%p)