#!/usr/bin/python3 # -*- coding: utf-8 -*- # Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, 134468. # Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, 66420. # We shall call a positive integer that is neither increasing nor decreasing a "bouncy" number; for example, 155349. # As n increases, the proportion of bouncy numbers below n increases such that there are only 12951 numbers below one-million that are not bouncy and only 277032 non-bouncy numbers below 10^10. # How many numbers below a googol (10^100) are not bouncy? from functions import nCr LIMIT = 10 print((nCr(LIMIT+10,10)-LIMIT-1)+(nCr(LIMIT+9,9)-1)-LIMIT*9)
#!/usr/bin/python3 # -*- coding: utf-8 -*- # 70 colored balls are placed in an urn, 10 for each of the seven rainbow colors. # What is the expected number of distinct colors in 20 randomly picked balls? # Give your answer with nine digits after the decimal point (a.bcdefghij). from functions import nCr print(str(7*(1-(nCr(60,20)/nCr(70,20))))[:11])
#!/usr/bin/python3 # -*- coding: utf-8 -*- from functions import nCr print(nCr(40, 20))
#!/usr/bin/python3 # -*- coding: utf-8 -*- # Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, 134468. # Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, 66420. # We shall call a positive integer that is neither increasing nor decreasing a "bouncy" number; for example, 155349. # As n increases, the proportion of bouncy numbers below n increases such that there are only 12951 numbers below one-million that are not bouncy and only 277032 non-bouncy numbers below 10^10. # How many numbers below a googol (10^100) are not bouncy? from functions import nCr LIMIT = 100 print((nCr(LIMIT+10,10)-LIMIT-1)+(nCr(LIMIT+9,9)-1)-LIMIT*9)
#!/usr/bin/python3 # -*- coding: utf-8 -*- # 70 colored balls are placed in an urn, 10 for each of the seven rainbow colors. # What is the expected number of distinct colors in 20 randomly picked balls? # Give your answer with nine digits after the decimal point (a.bcdefghij). from functions import nCr print(str(7 * (1 - (nCr(60, 20) / nCr(70, 20))))[:11])