Computer Language Symbols

Problem

A computer language is using 10 symbols and 26 letters. How many 6 digit words can be constructed using 3 different letters and 3 different symbols. How many different symbols can be constructed if the word has to start with letter Z for example.

Solution

For the first part of the question we need to choose 3 symbols from 10 and 3 letters from 26. This means

(10 C 3) x (26 C 3)

This gives us the total number of combinations but order does matter so we need to permute the 6 digits by multiplying the value above by 6 factorial. The total number of words is

(10 C 3) x (26 C 3) x 6! =
(10!/(7!x3!)) x (26!/(23!x3!)) x 6! =
224640000

For the second part of the question we need to choose 2 letters from 25 because the first letter has been already chosen for us which is z and we need to choose 3 symbols from 10. This means

(10 C 3) x (25 C 2)

Similarly since order is important we need to permute the individual digits by multiplying the number above by 5 factorial. Recall that the first character is fixed so instead of permuting 6 digits we permute 5 digits. The total number of words in this case is

(10 C 3) x (25 C 2) x 5! =
(10!/(7!x3!)) x (25!/(23!x2!)) x 5! =
4320000
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