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

Solution (Python):
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)

Solution (Python):
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)

Solution (Python):
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)

Solution (Python):
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)

Solution (Python):

n,k=map(int,input().split())
p=998244353
f=[1]

for i in range(k):
    
    f+=[(i+1)*f[i]%p]
    

print(sum(f[-2]*pow(f[i]*f[-2-i],-1,p)for i in range(n-1,k))%p)

Solution (Python):
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)

Solution (Python):
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)

Solution (Python):
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)

Solution (Python):
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)

Solution (Python):
def combs_mod(n,k,mod):
  
  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())

Solution (Python):
def combs_mod(n,k,mod):
  
  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())

Solution (Python):
def combs_mod(n,k,mod):
  
  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())

Solution (Python):
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)

Solution (Python):
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)

Solution (Python):
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)

Solution (Python):
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())))

Solution (Python):
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())))

Solution (Python):
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())))