Question

video lesson 5.5 example 1 – counting permutations
consider the permutations of the letters in the word pencils.

1. in how many ways can you arrange all the letters?
2. in how many ways can you arrange 3 of the letters?

Explanation:

Part 1: Arranging all letters of "PENCILS"

Step1: Check for repeated letters

The word "PENCILS" has 7 distinct letters (P, E, N, C, I, L, S). No repeated letters.

Step2: Use permutation formula for distinct objects

The number of permutations of \( n \) distinct objects is \( n! \) (n factorial). Here, \( n = 7 \), so we calculate \( 7! \).
\( 7! = 7\times6\times5\times4\times3\times2\times1 = 5040 \)

Step1: Identify the type of permutation

We need to find the number of permutations of 7 distinct letters taken 3 at a time. The formula for permutations of \( n \) objects taken \( r \) at a time is \( P(n, r)=\frac{n!}{(n - r)!} \). Here, \( n = 7 \) and \( r = 3 \).

Step2: Calculate the permutation

First, compute \( (n - r)!=(7 - 3)!=4! \). Then, \( P(7, 3)=\frac{7!}{4!}=\frac{7\times6\times5\times4!}{4!} \). The \( 4! \) terms cancel out, leaving \( 7\times6\times5 = 210 \).

Answer:

5040

Part 2: Arranging 3 of the letters of "PENCILS"