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

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

Step2: Calculate \(6!\)

\(6!=6\times5\times4\times3\times2\times1 = 720\)

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

The formula for permutations \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 15\) (number of young ladies) and \(r=3\) (number of positions - first three).

Step2: Calculate \(P(15,3)\)

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Step1: Permutation formula for circular permutations

The number of circular permutations of \(n\) distinct objects is \((n-1)!\). Here \(n = 9\) (number of students).

Step2: Calculate \((9 - 1)!\)

\((9-1)!=8!=8\times7\times6\times5\times4\times3\times2\times1=40320\)

Step1: Permutation formula for \(n\) objects with some non - distinct

The number of permutations of \(n\) objects where \(p_1\) of one kind, \(p_2\) of another kind,\(\cdots\), \(p_k\) of the \(k^{th}\) kind is \(\frac{n!}{p_1!p_2!\cdots p_k!}\). For the number \(3334777\), \(n=7\), \(p_1 = 3\) (number of \(3\)s), \(p_2=1\) (number of \(4\)s), \(p_3 = 3\) (number of \(7\)s)

Step2: Calculate the number of permutations

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Answer:

720