A company uses a coding system to identify its clients - NSC Mathematics - Question 11 - 2017 - Paper 1

To determine how many digits are needed to uniquely identify 700,000 clients, we can analyze using the following steps:

1. Calculating Total Combinations: If we use 2 letters and vary the number of digits in each code, the formulas are:

   $ext{Total combinations} = ( ext{ways to choose letters}) imes ( ext{ways to choose digits}).$

2. For 2 Digits:

   • Letters: $5 imes 5 = 25$
   • Digits: $10 imes 9 = 90$
   • Total: $25 imes 90 = 2250$

3. For 3 Digits:

   • Digits: $10 imes 9 imes 8 = 720$
   • Total: $25 imes 720 = 18000$

4. For 4 Digits:

   • Digits: $10 imes 9 imes 8 imes 7 = 5040$
   • Total: $25 imes 5040 = 126000$

5. For 5 Digits:

   • Digits: $10 imes 9 imes 8 imes 7 imes 6 = 30240$
   • Total: $25 imes 30240 = 756000$

Since 756,000 exceeds 700,000, a coding system with 5 digits will ensure each client can be uniquely identified.