How many three digit numbers can be formed from the digits 1, 2, 3, 4 and 5

I solved it this way because if a number needs to be formed with a certain number of digits (with no restrictions - which I think 'as often as desired' means), you assign a scale to the given constituent digits and then pick the same number of digits as the new number needs, going from highest to lowest. Then multiply these digits by each other to give the amount of different numbers that could be created.

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How many three digit numbers can be formed using the digits 1 2 3 4 5 6 if the digits cannot be repeated?

However my answer of 60 possible numbers conflicts with the textbook's answer of 64.

The next part of the question was finding how many 3-digit numbers can be formed using 2, 3, 4 and 5 using at most one each. I was able to get this question, by changing 2, 3, 4 and 5 to 1, 2, 3 and 4; then multiplying 4 by 3 by 2 to give 24 possibilities.

Hint: Fundamental principle of counting: According to the fundamental principle of counting if a task can be done in “m” ways and another task can be done in “n” ways, then the number of ways in which both the tasks can be done in mn ways.

Complete step by step solution:
Case [1]: Repetition not allowed:
The number of ways in which the ones place can be filled = 5 ways.
The number of ways in which the tens place can be filled = 4 ways (because repetition is not allowed. So, the choice of one place cannot be used).
The number of ways in which hundreds place can be filled = 3 ways.
Hence according to the fundamental principle of counting the number of ways in which the three places can be filled to form a three-digit number = $5\times 4\times 3=60$
Hence the number of 3-digit numbers formed using the digits 1, 2, 3, 4 and 5 = 60 i.e 5! ways
Case [2]: Repetition is allowed:
The number of ways in which the ones place can be filled = 5 ways.
The number of ways in which the tens place can be filled = 5 ways (because repetition is allowed. So, the choice of one's place can be used).
The number of ways in which hundreds place can be filled = 5 ways.
Hence according to the fundamental principle of counting the number of ways in which the three places can be filled to form a three-digit number = $5\times 5\times 5=125$
Hence the number of 3-digit numbers formed using the digits 1, 2, 3, 4 and 5 = 125

Note: The number of 3-digit numbers formed using the digits 1, 2, 3, 4 and 5 when repetition is not allowed is equivalent to the number of 3 Letter permutations of 5 distinct letter = ${}^{5}{{P}_{3}}=\dfrac{5!}{\left( 5-3 \right)!}=\dfrac{5!}{2!}=\dfrac{120}{2}=60$ which is the same result as above.

Ex 7.1, 1 How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that (i) repetition of the digits is allowed? 3 digit number : Number of 3 digit numbers with repetition = 5 × 5 × 5 = 125

Solution : (i) When repetition of digits is allowed:
No. of ways of choosing firsy digits = 5
No. of ways of choosing second digit = 5
No. of ways of choosing third digit = 5
Therefore, total possible numbers `= 5 xx 5 xx 5 = 125`
(ii) When repetition of digits is not allowed:
No. of ways of choosing first digit = 5
No. of ways of choosing second digit = 4
No. of ways of choosing thrid digit = 3
Total possible numbers `= 5 xx 4 xx 3 = 60`.

How many seven-digit phone numbers can be formed if the first digit cannot be 0 and repetition of digits is not permitted?

Statistics Probability Combinations and Permutations

1 Answer

BeeFree

Jan 28, 2016

Take it step-by-step ...

Explanation:

There are 9 ways to pick the first digit [1-9]

Next, there are 9 ways to pick the second digit [0 plus the remaining 8 from the first digit]

The 3rd digit is chosen from the remaining 8 numbers.

The 4th digit is chosen from the remaining 7 numbers.

The 5th digit is chosen from the remaining 6 numbers.

The 6th digit is chosen from the remaining 5 numbers.

The 7th digit is chosen from the remaining 4 numbers.

So, in summary, there are ...

#9xx9xx8xx7xx6xx5xx4#ways#=544320#ways

hope that helped

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How many 3

Thus, The total number of 3-digit numbers that can be formed = 5×4×3 = 60.

How many three digit numbers can be formed using the digits 1 2 3 4 5 6 if the digits cannot be repeated?

Therefore 120 such numbers are possible.

How many 3

If repetition of digits is allowed: 7*7*7 = 7^3 = 343 three digit numbers can be formed. If repetition of digits is not allowed: 7*6*5 = 210 three digit numbers can be formed.

How many 3

There are 9 digits available to form 3-digits numbers without repetition. There are 9 choices for the hundreds digit, 8 choices for the tens digit, and 7 choices for the units digit. Therefore, 9 * 8 * 7 = 504 such 3-digits are possible.

How many 3