Why does 2^6-1 equal the number of Braille symbols?

[roadrunner]

permutations/combinations help please

Hello everyone im' stuck on this problem.

It says:
Each symbol in braille code is represened by a rectangular arrangement of six dots. Given that a least 1 dot of the 6 must be raised, how many symbols can be represented in brail?

now i saw this posted somewhere else, they got 63. don't know how.

i got 63 two ways...

2^6-1 (case where all down)=63

6c1+6c2+6c3+6c4+6c5+6c6 -1 also =63 (where 6c3 etc is combinations...6 options choose 3)

can sumone explain how/why that works please.

also part b...how many combinations have EXACTLY 3 raised

and how many have an even number of raised dots

 

[CRGreathouse]

Hello everyone im' stuck on this problem.

How are you stuck? You gave two good answers to the question, both of which are correct and correctly explained. The second approach gives a direct answer to your other questions -- 6 choose 3 ways to have three raised, and the appropriate combinations for only even ones raised.

 

[roadrunner]

im stuck on part b lol exactly 3 rasied
is is 6c3=20?
and part c...how many symbols have an even number of raised dots...
so 6c2+6c4+6c6?