Question

lesson 4 practice: permutations with restrictions

1. how many arrangements can be made of the word

a) matrix if m is first?

b) minus if n is first and s is last?

c) absolute if a consonant is last?
d) factor if the vowels are first and last?

Explanation:

Part (a): Word "MATRIX" with M first

Step1: Fix M at first position

Since M is fixed in the first spot, we now need to arrange the remaining 5 letters (A, T, R, I, X). The number of permutations of \( n \) distinct objects is \( n! \). Here, \( n = 5 \).

Step2: Calculate permutations of remaining letters

The number of permutations of 5 distinct letters is \( 5! \). We know that \( n! = n\times(n - 1)\times(n - 2)\times\cdots\times1 \), so \( 5! = 5\times4\times3\times2\times1 = 120 \).

Step1: Fix N at first and S at last

We fix N in the first position and S in the last position. Now we need to arrange the remaining 3 letters (M, I, U) in the middle 3 positions.

Step2: Calculate permutations of remaining letters

The number of permutations of 3 distinct letters is \( 3! \). Using the formula \( n! = n\times(n - 1)\times(n - 2)\times\cdots\times1 \), we get \( 3! = 3\times2\times1 = 6 \).

Step1: Identify vowels and consonants

In the word "ABSOLUTE", the letters are: A (vowel), B (consonant), S (consonant), O (vowel), L (consonant), U (vowel), T (consonant), E (vowel). So vowels: A, O, U, E (4 vowels); consonants: B, S, L, T (4 consonants). We need the last letter to be a consonant.

Step2: Choose last letter (consonant)

We have 4 choices for the last position (any of the 4 consonants).

Step3: Arrange remaining letters

After choosing the last letter, we have 7 remaining letters to arrange in the first 7 positions. The number of permutations of 7 distinct letters is \( 7! \).

Step4: Calculate total arrangements

Using the multiplication principle, total arrangements = number of choices for last letter × permutations of remaining letters. So total arrangements = \( 4\times7! \). Calculate \( 7! = 7\times6\times5\times4\times3\times2\times1 = 5040 \), then \( 4\times5040 = 20160 \).

Answer:

The number of arrangements is \( 120 \).

Part (b): Word "MINUS" with N first and S last