Question

How many 4 digit numbers (with possible repetition) can be formed with no digit less than 4​

Answer 1

Answer:

Hence total number of permutations = 9×504=4536.

Step-by-step explanation:

Step : 1 A four digit number is to be formed from the digits 0,1,2,3,4,5,6,7,8,9.

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Since the thousandth slot, which is on the left, cannot contain zero, There are nine different methods to fill a thousand.

Since duplication is prohibited, the remaining 9 digits can occupy the hundreds position.

So there are nine possibilities to fill the position of hundred.

Step : 2  Similarly, there are 8 digits left to complete the tens slot.

Therefore, there are eight different ways to fill tens of places.

Similarly, we still have 7 numbers to complete one slot.

Thus, there are seven methods to fill a space.

Therefore, the needed number of ways to create four-digit numbers with the specified digits is 9x9x8x7=4536.

Step : 3  Alternative Method: Since the digit 0 cannot be used, any of the digits from 1 to 9 should be used to fill in the thousands place of the 4-digit number. Consequently, there are nine different ways that a thousand spots can be filled.

Any number between 0 and 9 can be used to represent hundreds, tens, and units.

Step : 4  The numerals, however, cannot be repeated. A digit has already taken up space in 4-digit numerals and thousands. The remaining 9 digits must go in the hundreds, tens, and unit places.

Step : 5 There will be as many of these three-digit numbers as there are combinations of the nine distinct digits when they are taken three at a time.

${ }^9 \mathrm{P}_3=\frac{9 !}{(9-3) !}=\frac{9 !}{6 !}$

Number of such 3-digit numbers

$\frac{9 \times 8 \times 7 \times 6 !}{6 !}=9 \times 8 \times 7=504$

Thus, by multiplication principle, the required number of 4-digit numbers is 9×504=4536

Answer 2

There are 625 four digit numbers (with possible repetition) that can be formed with no digit less than 4​.

• The number of possible arrangements for a given set is calculated mathematically, and this process is known as permutation. The term permutation refers to the variety of possible arrangements or orders. The arrangement's order is important when using permutations. Permutations come in three varieties, one without repetition and two with repetition.
• They stand for a spectrum of group configurations that are possible. When it comes to permutations, order is crucial. This distinguishes it from a combination, a term in which sequence is irrelevant. Permutations are a type of ordered combination to some extent.

Here, according to the given information, we are given that,

We need to form 4 digit numbers which will contain repetition of digits. Also, none of the digits should be less than 4. Then, the digits that are possible to be placed in the number are 5, 6, 7, 8 and 9.

This means that, the first place has a probability of having any of the 5 digits. The next place will also have a probability of having any of the 5 digits. Similarly, the third and the fourth places will also have a probability of having any of the 5 digits.

This means that total number 4 digits numbers to be formed in this manner = 5 × 5 × 5 × 5 = 625.

Hence, there are 625 four digit numbers (with possible repetition) that can be formed with no digit less than 4​.

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