Question

Two spies have to communicate using secret code. They need to create exactly 30 possible precoded messages, using a single number and letter. Which structure should the code have? A. Select a number from {1, 2, 3, 4} and a vowel
B. Select a number from {1, 2, 3, 4, 5} and a vowel.

C. Select a number from {1, 2, 3, 4, 5, 6} and a vowel.

D. Select a number from {1, 2, 3, 4, 5} and a consonant

Answer 1

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The structure should the code have should be letter C which is Select a number from {1, 2, 3, 4, 5, 6} and a vowel.

Answer 2

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.