A binary number is a number that consists of only the digits 0 or 1.

Question

A binary number is a number that consists of only the digits 0 or 1. How many distinguishable binary numbers can be formed from the binary number 1111100011111000?
Use all of the digits, realizing that the number must begin with 1 in order to use all of the digits.

Answer 1

the first digit must be a 1, so you have 10 choices

for the second digit you have 15 digits to choose from (you used one of the 1's, leaving 15 digits)

for the third digit you have 14 digits to choose from

and so on...

10*15! so far but there are 10 1's and 6 0's

the 1's are indistinguishable and the 0's are indistinguishable

therefore divide 10*15! by 10! and also by 6!

(10*15!)/(10!*6!)

(10*15*14*13*12*11)/(6*5*4*3*2*1)   (the two 10!'s cancel from the numerator and denominator)

simplify or multiply as is

after simplifying you get 5*7*13*11

5*7*13*11=5005

if you do all the multiplying you get 3,603,600/720=5005