Help?
A bank assigns a personal code to each of its customers. This bank currently has 1,207,354 customers. The personal code uses 2 distinct letters from 24 letters (all the letters in the alphabet except O and I), followed by n distinct numerals from 0 to 9.
What is the smallest value of n such that each customer gets a unique code?
1207354 ≤ (24 * 24 * 10^n)
1207354 / 576 ≤ 10^n
log(1207354 / 576) ≤ n
n distinct digits means 10Pn = 10!/(10-n)! not 10^n
So adjust the calculation as needed
Of course, you know that n <= 10
missed the "distinct" ... no repetitions
24 * 23 ... not 24^2
(10-n)! ≤ 10! * 552 / 1207354
To find the smallest value of n such that each customer gets a unique code, we need to consider the number of possible combinations for the letters and numerals.
First, let's focus on the letters. The bank uses 2 distinct letters from 24 letters, which means we need to calculate the number of combinations of 2 letters from a set of 24. This can be done using the formula for combinations, which is calculated as nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items chosen at a time.
In this case, we have n = 24 (the total number of letters) and r = 2 (the number of letters chosen). Plugging these values into the formula, we get:
24C2 = 24! / (2!(24-2)!) = 24! / (2!22!)
Now, let's calculate the number of combinations for the numerals. Since we need to choose n distinct numerals, we simply have to count the number of digits from 0 to 9, which is 10.
Now we can calculate the total number of unique codes possible by multiplying the number of combinations for the letters and numerals:
Total number of unique codes = 24C2 * 10^n
To find the smallest value of n, we want the total number of unique codes to be equal to or greater than the number of customers (1,207,354). Therefore, we can set up the equation:
24C2 * 10^n >= 1,207,354
Now, solving this equation for the smallest value of n requires some trial and error. We can start with n = 1 and calculate the left-hand side of the equation until we find the smallest value of n that satisfies the inequality. Here's a breakdown of the calculations:
n = 1: 24C2 * 10^1 = 24 * 10 = 240 (too small)
n = 2: 24C2 * 10^2 = 24 * 100 = 2400 (too small)
n = 3: 24C2 * 10^3 = 24 * 1000 = 24,000 (too small)
n = 4: 24C2 * 10^4 = 24 * 10,000 = 240,000 (too small)
n = 5: 24C2 * 10^5 = 24 * 100,000 = 2,400,000 (greater than 1,207,354)
Therefore, the smallest value of n such that each customer gets a unique code is 5.