Answer:
The code should have the structure C.
Step-by-step explanation:
Given : Two spies have to communicate using secret code. They need to create exactly 30 possible precoded messages, using a single number and letter.
To Find: Which structure should the code have?
Solution:
The structure that have 30 possible outcomes , the code should have that structure .
Option A: Select a number from {1, 2, 3, 4} and a vowel
Since we are given that the code contains one number and one letter
No.of vowels = {a,e,i,o,u}=5
Out of these five we will choose only one
We are supposed to choose a number from {1, 2, 3, 4}
Out of these four numbers we will choose only one
Now to find no. of possible outcomes we will use combination
Formula : \(^nC_r=\frac{n!}{r!(n-r)!}\)
So, no. of possible outcomes from option A :
\(^5C_1\times ^4C_1\)
\(\frac{5!}{1!(5-1)!} \times \frac{4!}{1!(4-1)!}\)
\(\frac{5!}{1!(4)!} \times \frac{4!}{1!(3)!}\)
\(5 \times 4\)
\(20\)
Thus no. of possible outcomes from Option A is 20.
Option B: Select a number from {1, 2, 3, 4, 5} and a vowel.
Since we are given that the code contains one number and one letter
No.of vowels = {a,e,i,o,u}=5
Out of these five we will choose only one
We are supposed to choose a number from {1, 2, 3, 4,5}
Out of these five numbers we will choose only one
So, no. of possible outcomes from option B :
\(^5C_1\times ^5C_1\)
\(\frac{5!}{1!(5-1)!} \times \frac{5!}{1!(5-1)!}\)
\(\frac{5!}{1!(4)!} \times \frac{5!}{1!(4)!}\)
\(5 \times 5\)
\(25\)
Thus no. of possible outcomes from Option B is 25.
Option C: Select a number from {1, 2, 3, 4, 5, 6} and a vowel.
Since we are given that the code contains one number and one letter
No.of vowels = {a,e,i,o,u}=5
Out of these five we will choose only one
We are supposed to choose a number from {1, 2, 3, 4,5,6}
Out of these six numbers we will choose only one
So, no. of possible outcomes from option C :
\(^5C_1\times ^6C_1\)
\(\frac{5!}{1!(5-1)!} \times \frac{6!}{1!(6-1)!}\)
\(\frac{5!}{1!(4)!} \times \frac{6!}{1!(5)!}\)
\(5 \times 6\)
\(30\)
Thus no. of possible outcomes from Option C is 30.
So, Option C is correct
Option D: Select a number from {1, 2, 3, 4, 5} and a consonant
Since we are given that the code contains one number and one letter
No.of consonants = 21
Out of these twenty one we will choose only one
We are supposed to choose a number from {1, 2, 3, 4,5}
Out of these five numbers we will choose only one
So, no. of possible outcomes from option C :
\(^5C_1\times ^21C_1\)
\(\frac{5!}{1!(5-1)!} \times \frac{21!}{1!(21-1)!}\)
\(\frac{5!}{1!(4)!} \times \frac{21!}{1!(20)!}\)
\(5 \times 21\)
\(105\)
Thus no. of possible outcomes from Option D is 105.
Hence Option C is correct because no. of possible outcomes in that case is 30 .
So, the code should have the structure C.
