Question Video: Finding the Number of Ways to Arrange a Given Set of Digits to Form an 𝑥-Digit Number with a Given Criteria | Nagwa Question Video: Finding the Number of Ways to Arrange a Given Set of Digits to Form an 𝑥-Digit Number with a Given Criteria | Nagwa

# Question Video: Finding the Number of Ways to Arrange a Given Set of Digits to Form an 𝑥-Digit Number with a Given Criteria Mathematics • Third Year of Secondary School

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In how many ways can an odd number of 6 digits be formed using the numbers 1, 2, 3, 4, 5, 6 if no digits are to be repeated?

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### Video Transcript

In how many ways can an odd number of six digits be formed using the numbers one, two, three, four, five, six if no digits are to be repeated?

To answer this question, we’re going to use something called the product rule for counting. The product rule for counting can save us a little bit of time because we don’t need to list out every possible number we can make. And it says to find the total number of outcomes for two or more events, we multiply the number of outcomes for each event together. Here we have six digits. And we’re looking to make an odd number. Well, an odd number will end in the digits one, three, or five. And so we’ll just begin by working out how many ways there are for us to make an odd number.

Let’s imagine we’re choosing the final digit of our six-digit number from this list. We begin by picking that final digit, and so there are three ways to choose that digit. It can be a one, a three, or a five. Since we’ve already picked a digit, we know that the next digit, the second digit in our six-digit number, will be chosen out of the remaining five. Then there are only four digits left. So there are four ways to choose the third digit. We then choose the fourth digit.

Well, there are only three numbers left to choose from. So there are three ways to choose this number. There are two ways to choose the fifth number. And then we have no choice on the sixth number. There is just one way to choose that. And so the number of ways of choosing an odd number of six digits from our list, assuming that none are repeated, is three times five times four times three times two times one, which is equal to 360. There are 360 ways to choose an odd number using our six digits.

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