def compute():
square_free = set()
for n in range(1, 51):
for k in range(int(n/2)+1):
if all(choose(n,k) % (p**2) != 0 for p in prime_factors):
square_free.add(choose(n,k))
return sum(square_free)
def only_red_and_orange(n):
if n == min_num_colors:
return 1
return choose(n * balls_per_color, balls_to_draw) - sum([
choose(n, x) * only_red_and_orange(x)
for x in xrange(min_num_colors, n)
])
def run():
total = 0
for n in range(23, 101):
subtotal = 0
for r in range(1, n):
if euler.choose(n, r) >= 1000000:
break
subtotal += 1
total += n - 2*subtotal - 1
return total
def naive_transition_matrix( n ):
states = [ tuple(state) for state in euler.permutations( range(1,n+1) ) ]
m = len(states)
mat = numpy.zeros( (m,m) )
state2id = dict(zip(states, range(len(states))))
id2state = dict(zip(range(len(states)), states))
choices = list(euler.choose(range(n), 2))
for u in states[1:]:
for i, j in choices:
v = list(u)
v[i], v[j] = v[j], v[i]
v = tuple(v)
mat[ state2id[u], state2id[v] ] = 1.0/len(choices)
return numpy.matrix(mat)
def run():
primes = euler.primesUpTo(18097)
pascals = set()
for n in range(0, 50):
for k in range(0, n//2+1):
pascals.add(euler.choose(n, k))
total = 0
for pascal in pascals:
squarefree = True
for prime in primes:
if pascal % (prime*prime) == 0:
squarefree = False
break
if squarefree:
total += pascal
return total
def only_red_and_orange(n):
if n == min_num_colors:
return 1
return choose(n * balls_per_color, balls_to_draw) - sum([choose(n, x) * only_red_and_orange(x) for x in xrange(min_num_colors, n)])
from euler import choose
colors_in_rainbow = 7
balls_to_draw = 20
balls_per_color = 10
total_balls = balls_per_color * colors_in_rainbow
min_num_colors = balls_to_draw / balls_per_color
# How many ways can we choose balls_to_draw balls from our collection,
# selecting from exactly n _specified_ colors
def only_red_and_orange(n):
if n == min_num_colors:
return 1
return choose(n * balls_per_color, balls_to_draw) - sum([choose(n, x) * only_red_and_orange(x) for x in xrange(min_num_colors, n)])
def only_n_distinct_colors(n):
return choose(colors_in_rainbow, n) * only_red_and_orange(n)
print float(sum([a * only_n_distinct_colors(a) for a in range(min_num_colors, colors_in_rainbow + 1)])) / choose(total_balls, balls_to_draw)
]
def constr(a, b, v1, v2):
if a in v1 and b in v2:
return True
def row_or(row, v1, v2):
for a, b in row:
if constr(a, b, v1, v2):
return True
return False
def col_and(cols, v1, v2):
for row in cols:
if row_or(row, v1, v2) == False:
return False
return True
combs = set()
faces = list(euler.choose(range(10), 6))
for d1 in faces:
for d2 in faces:
if col_and(lst, d1, d2):
a = tuple(d1)
b = tuple(d2)
ab = (a, b)
if b < a:
ab = (b, a)
combs.add(ab)
print len(combs)
import euler size = 20 print euler.choose(2*size, size)
def euler53():
print(sum(1 for n in range(1, 101) for r in range(n) if choose(n, r) > 1000000))
def euler15():
grid_size = 20
print(choose(grid_size * 2, grid_size))
def non_bouncy(n):
return choose(n+10, n) + choose(n+9, 9) - 10*n - 2
from decimal import Decimal
from euler import choose,product
ways = [Decimal(0) for _ in range(0,8)]
for r in range(0,11):
for o in range(0,min(11, 21-r)):
for y in range(0,min(11,21-r-o)):
for g in range(0,min(11,21-r-o-y)):
for b in range(0,min(11,21-r-o-y-g)):
for i in range(0,min(11,21-r-o-y-g-b)):
v = 20 - sum([r,o,y,g,b,i])
roygbiv = [r,o,y,g,b,i,v]
count = sum(1 for x in roygbiv if x > 0)
different_ways = product(choose(10,x) for x in roygbiv)
ways[count] += Decimal(different_ways)
tot_ways = sum(ways)
EV = sum(Decimal(x) * (ways[x] / tot_ways) for x in range(2,8))
print(EV)
def compute():
return len([(n,r) for n in range(101) for r in range(1,n) if choose(n,r) > 10**6])
def only_n_distinct_colors(n):
return choose(colors_in_rainbow, n) * only_red_and_orange(n)
from euler import choose print choose(40, 20)
from euler import choose
colors_in_rainbow = 7
balls_to_draw = 20
balls_per_color = 10
total_balls = balls_per_color * colors_in_rainbow
min_num_colors = balls_to_draw / balls_per_color
# How many ways can we choose balls_to_draw balls from our collection,
# selecting from exactly n _specified_ colors
def only_red_and_orange(n):
if n == min_num_colors:
return 1
return choose(n * balls_per_color, balls_to_draw) - sum([
choose(n, x) * only_red_and_orange(x)
for x in xrange(min_num_colors, n)
])
def only_n_distinct_colors(n):
return choose(colors_in_rainbow, n) * only_red_and_orange(n)
print float(
sum([
a * only_n_distinct_colors(a)
for a in range(min_num_colors, colors_in_rainbow + 1)
])) / choose(total_balls, balls_to_draw)
if a in v1 and b in v2:
return True
def row_or(row, v1, v2):
for a, b in row:
if constr(a, b, v1, v2):
return True
return False
def col_and(cols, v1, v2):
for row in cols:
if row_or(row, v1, v2) == False:
return False
return True
combs = set()
faces = list(euler.choose(range(10), 6))
for d1 in faces:
for d2 in faces:
if col_and(lst, d1, d2):
a = tuple(d1)
b = tuple(d2)
ab = (a, b)
if b < a:
ab = (b, a)
combs.add(ab)
print len(combs)
#!/usr/bin/env python """ Problem 015: Starting in the top left corner of a 2 x 2 grid, there are 6 routes (without backtracking) to the bottom right corner. How many routes are there through a 20 x 20 grid? """ from euler import choose N = 20 print(choose(2 * N, N)) # Central binomial coefficient
def only_n_distinct_colors(n):
return choose(colors_in_rainbow, n) * only_red_and_orange(n)
def run():
total = 0
depth = 100
for n in range(1, 10):
total += euler.choose(depth+n-1, n) + euler.choose(depth+n, n+1)
return total-9*depth
SIZE = 20
grid = [[1 for _ in range(SIZE + 1)] for _ in range(SIZE + 1)]
for i in range(2, (SIZE + 1) * 2):
for x in range(1, i):
y = i - x
if x >= SIZE + 1 or y >= SIZE + 1:
continue
grid[x][y] = grid[x - 1][y] + grid[x][y - 1]
print grid[SIZE][SIZE]
###
# C(40, 20) = 20! / 40!(40-20)!
import euler
print euler.choose(SIZE * 2, SIZE)
def euler53():
print(
sum(1 for n in range(1, 101) for r in range(n)
if choose(n, r) > 1000000))
def EN(k):
if k == 0:
return rational(0)
return rational(1,(2**k-1)) * sum(
[choose(k, i,1) * (EN(k - i) + 1) for i in range(1,k + 1)], rational(1))
def compute():
return 7.0 * (1 - 1.0 * choose(60, 40) / choose(70, 50))
import euler size = 20 print euler.choose(2 * size, size)
from euler import choose, squarefree, primes
N = 51
ps = primes(int(choose(N-1,(N-1)//2)**.5))
print("got primes")
sfrees = set()
for n in range(0,N):
for k in range(0,(n+3)//2):
coeff = choose(n,k)
if squarefree(coeff,ps):
sfrees.add(coeff)
print(sum(sfrees))
def run():
return euler.choose(40, 20)