Questions: Suppose we want to choose 7 objects, without replacement, from 10 distinct objects. (If necessary, consult a list of formulas.) (a) How many ways can this be done, if the order of the choices is relevant? (b) How many ways can this be done, if the order of the choices is not relevant?

Suppose we want to choose 7 objects, without replacement, from 10 distinct objects. (If necessary, consult a list of formulas.) (a) How many ways can this be done, if the order of the choices is relevant? (b) How many ways can this be done, if the order of the choices is not relevant?
Transcript text: Suppose we want to choose 7 objects, without replacement, from 10 distinct objects. (If necessary, consult a list of formulas.) (a) How many ways can this be done, if the order of the choices is relevant? (b) How many ways can this be done, if the order of the choices is not relevant?
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Solution

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Solution Steps

Step 1: Understanding the Problem

We are given a problem where we need to select \(r=7\) objects from \(n=10\) distinct objects with order being relevant.

Step 2: Applying the Permutation Formula

To solve this, we use the permutation formula: \[P(n, r) = \frac{n!}{(n-r)!}\]

Step 3: Calculation

Substituting the given values into the formula, we get \[P(10, 7) = \frac{10!}{10-7!} = 604800\]

Final Answer:

The number of ways to select \(r=7\) objects from \(n=10\) distinct objects with order being relevant is 604800.

Step 1: Understanding the Problem

We are given a problem where we need to select \(r=7\) objects from \(n=10\) distinct objects without considering the order.

Step 2: Applying the Combination Formula

To solve this, we use the combination formula: \[C(n, r) = \frac{n!}{r!(n-r)!}\]

Step 3: Calculation

Substituting the given values into the formula, we get \[C(10, 7) = \frac{10!}{7!(10-7)!} = 120\]

Final Answer:

The number of ways to select \(r=7\) objects from \(n=10\) distinct objects without considering the order is 120.

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