## Thursday, March 15, 2012

Question:
If a person dials a sequence of numbers on the telephone, what possible words/strings can be formed from the letters associated with those numbers?

Let's analyze this.

On a handset, there are keys with number '0' to '9'.  Assume we have to key in full phone number (area code + number),  it's gonna be 10 digits.  Only certain keys have alphabets: 2 [ABC], 3 [DEF], 4 [GHI], 5 [JKL], 6 [MNO], 7 [PQRS], 8 [WXYZ], with total number of sets = 8. The question is to find what is the total number of permutation of these 10-digits out of n set of possible phone numbers

Using permutation formula:

 Formula: Note: , where nPr is the formula for permutations of n objects taken r at a time.

The set of possible numbers then is: [222-222-2222 to 888-888-8888] or 6,666,666,666 possible phone numbers.

We then compute:
n = 10,000,000,000
r = 10

Permutation =6,666,666,666 !/(10!(6,666,666,666  - 10)!) = (6,666,666,656 * 6,666,666,657 * 6,666,666,658 *6,666,666,669 * 6,666,666,660 * 6,666,666,661 * 6,666,666,662 * 6,666,666,663 * 6,666,666,664 * 6,666,666,665 * 6,666,666,666)/3628800 = <just figure out your self!>

If the allowed valid number is only 1 digit, then computation would be easier:

Permutation = 6,666,666,666!/(1!*(6,666,666,666-1)!) = 6,666,666,666 possible words