Can someone give me a formula to calculate $ _nP_0 + _nP_1 +_ nP_2 + _nP_3 + .. + _nP_x$ ?
I need a simple formula to calculate this.
MY actual question is
In a certain programming language identifiers must meet the following requirements: - the first character must be an ASCII letter (A-Z, a-z) - other (than the first) characters must be an ASCII letter, digit (0-9) or an underscore - the maximum length of an identifier is 8 characters The total number of possible identifiers is:
[A] $53*(63^7)$
[B] $52*(63^8 - 1)/62$
[C] $52*(63^7 - 1)/63$
[D] $53*(63^8)$
[E] $(63^8)/(52^8)$
Choose the right answer.
For 1 letter variable ->
Here, First position can be filled with 52 chars.
For 2 letter variable ->
Then, First posision can be filled with 52 chars. And second char can be filled up $_{63}P_1$
For 3 letter variable ->
Then, First posision can be filled with 52 chars. And second char can be filled up $_{63}P_1$. And third char can be filled up with $_{63}P_2$
This goes on like this upto 8 letter variable.
So, the series should be $52 + 52(_{63}P_1) + 52(_{63}P_2) + \dots $
That implies $52(1 + _{63}P_1 + _{63}P_2 + _{63}P_3 +\dots + _{63}P_7)$
After this, I couldn't proceed.