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How many distinct 5-letter code words can be formed from the letters in "TANTE" if letters may not be repeated and if letters may be repeated?

Mathematics

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Answers

2 parts
i
60
ii
1024

Step-by-step solution

7 steps

Here's how to solve the problem about forming code words from the letters in "TANTE":

First, let's list the letters in the word "TANTE": T, A, N, T, E.
There are 5 letters in total.
The distinct letters are T, A, N, E (4 distinct letters).
The letter 'T' appears twice.

Part i· 3 steps

The letters may not be repeated?

This means we are arranging all 5 letters from the word "TANTE" to form a 5-letter code word. Since the letter 'T' is repeated twice, we use the formula for permutations with repetitions.

  1. Identify the total number of letters and the count of repeated letters.

    Total number of letters (nn) = 5 (T, A, N, T, E).
    The letter 'T' is repeated (nTn_T) = 2 times.

  2. Apply the permutation formula for objects with repetitions.

    The number of distinct permutations is given by n!nT!\frac{n!}{n_T!}.

    Number of code words=5!2!\text{Number of code words} = \frac{5!}{2!}

  3. Calculate the factorials and simplify.

    5!2!=5×4×3×2×12×1\frac{5!}{2!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} =5×4×3= 5 \times 4 \times 3 =60= 60

    There are 60\boxed{60} different 5-letter code words if the letters may not be repeated (meaning each instance of a letter from "TANTE" is used exactly once).

Part ii1024

The letters may be repeated?

This means we can choose any of the distinct letters from "TANTE" for each position in the 5-letter code word, and we can use them multiple times.

  1. Identify the number of distinct letters available.

    The distinct letters in "TANTE" are T, A, N, E. There are 4 distinct letters.

  2. Determine the number of positions in the code word.

    We are forming a 5-letter code word, so there are 5 positions.

  3. Calculate the number of code words with repetition.

    Since repetition is allowed, for each of the 5 positions, there are 4 choices (T, A, N, E).

    Number of code words=(Number of distinct letters)Number of positions\text{Number of code words} = (\text{Number of distinct letters})^{\text{Number of positions}} =45= 4^5

  4. Calculate the value.

    45=4×4×4×4×44^5 = 4 \times 4 \times 4 \times 4 \times 4 =1024= 1024

    There are 1024\boxed{1024} different 5-letter code words if the letters may be repeated.

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