[Math] possible number of PAN card numbers

combinationspermutations

Permanent Account Number (PAN) is a code that acts as identification of Indians. An example number would be in the form of AAAPL1234C. The format is

  • The first three letters are sequence of alphabets from AAA to ZZZ
  • The fourth character informs about the type of holder of the card (possible letters A, B, C, F, G, H, L, J, P, T, K)
  • The fifth character of the PAN is the first character
    • of the surname or last name of the person, in the case of a "Personal" PAN card
    • of the name of the Entity, Trust, society, or organisation in the case of Company/ HUF/ Firm/ AOP/ BOI/ Local Authority/ Artificial Judicial Person/ Govt
  • next four numbers are ranging from 0001 to 9999
  • The last character is an alphabetic check digit

reference: wiki

Now, I'm trying to find the total possible number of PAN numbers like below

  1. for first 3 alphabets – 26 * 26 * 26
  2. fourth letter have only 11 possiblities – 11
  3. fifth letter can be any of the 26 alphabets – 26
  4. next 4 letters are between 0001 to 9999 – 9999
  5. last letter can be any of the 26 aplhabets – 26

so total would be 26 * 26 * 26 * 11 * 26 * 9999 * 26.

is this correct or am I missing something?

Best Answer

If everything you say is true, then yes you are correct. I just have concerns about items 4. and 5.

  1. You are correct if these four numbers are any numbers $0001$ to $9999$. There are $9999$ possible numbers. However, it is possible that the first three digits are any value $0$ to $9$ and the last digit is any value $1$ to $9$. In this case there are $10\times10\times10\times9 = 9000$ ways to fill out these four digits. This case excludes all values that end in $0$, like $10, 100, 1000$, etc.

  2. It is possible that there is some algorithm to calculate this "alphabetic check", so it might not be true that all 26 letters are possible.

I do not believe the article addressed these two items. If we simply ignore them and assume that your understanding of the structure is correct, then your calculation is correct.

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