Question

iii. what can i do with what i know: solve the following problems. write your solutions and answers on a separate sheet of paper. (3 points each) 1. how many ways can 6 students be seated in a row of 6 chairs for pictorial? 2. if fifteen young ladies compete for the first three most beautiful ladies in a pageant, in how many ways can the winners be selected assuming that there are no ties? 3. nine students are to be seated in a row of chairs. how many arrangements are possible if 3 students insist on sitting beside each other? 4. how many different numbers can be formed using all the digits of 3 3 4 7 7 7? 5. in how many ways can 9 different colored beads be arranged on a bracelet?

Explanation:

Step1: Permutation formula

For \(n\) distinct objects arranged in \(n\) positions, the number of permutations is \(n!\). Here \(n = 6\) (6 students and 6 chairs). The formula for permutations of \(n\) objects taken \(n\) at a time is \(P(n,n)=\frac{n!}{(n - n)!}=n!\).

$$6! = 6\times5\times4\times3\times2\times1$$

Step2: Calculate the factorial

$$6\times5\times4\times3\times2\times1=720$$

Step1: Permutation formula for \(n\) objects taken \(r\) at a time

The formula for permutations of \(n\) objects taken \(r\) at a time is \(P(n,r)=\frac{n!}{(n - r)!}\). Here \(n = 15\) (15 young ladies) and \(r=3\) (selecting 3 winners).

$$P(15,3)=\frac{15!}{(15 - 3)!}=\frac{15!}{12!}$$

Step2: Simplify the factorial expression

\(\frac{15!}{12!}=15\times14\times13\) (since \(n!=\ n\times(n - 1)\times\cdots\times(n - r+1)\times(n - r)!\))

$$15\times14\times13 = 2730$$

Step1: Permutation formula for \(n\) objects taken \(r\) at a time

The formula for permutations of \(n\) objects taken \(r\) at a time is \(P(n,r)=\frac{n!}{(n - r)!}\). Here \(n = 9\) (9 students) and \(r = 3\) (3 students to be seated).

$$P(9,3)=\frac{9!}{(9 - 3)!}=\frac{9!}{6!}$$

Step2: Simplify the factorial expression

\(\frac{9!}{6!}=9\times8\times7\) (since \(n!=\ n\times(n - 1)\times\cdots\times(n - r+1)\times(n - r)!\))

$$9\times8\times7=504$$

Step1: Analyze the number of digits

The number \(3334777\) has \(7\) digits. The digit \(3\) appears \(3\) times, the digit \(4\) appears \(1\) time and the digit \(7\) appears \(3\) times.
The formula for permutations of a multi - set with \(n\) objects, where there are \(n_1\) of one kind, \(n_2\) of another kind,\(\cdots\), \(n_k\) of the \(k\)th kind is \(\frac{n!}{n_1!n_2!\cdots n_k!}\). Here \(n=7\), \(n_1 = 3\) (for digit \(3\)), \(n_2=1\) (for digit \(4\)) and \(n_3 = 3\) (for digit \(7\)).

$$N=\frac{7!}{3!1!3!}$$

Step2: Calculate the factorial values

$$7! = 5040$$

, \(3! = 6\), \(1! = 1\)

$$N=\frac{5040}{6\times1\times6}=\frac{5040}{36}=140$$

Answer:

720