Class 11

Math

Algebra

Permutations and Combinations

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?

ii) Repetition of the digits is not allowed?

A three digit number is to be formed from given 5 digits $1,2,3,4,5$.

$□□□HTO$

Now, there are $5$ ways to fill ones place.

Since, repetition is allowed , so tens place can also be filled by $5$ ways.

Similarly,hundreds place can also be filled by $5$ ways.

So, number of ways in which three digit numbers can be formed from the given digits is $5×5×5=125$

**Alternative Method:**

There will be as many way ways as there are ways of filling $3$ vacant places in succession by the given five digits. In this case, repetition of digits is allowed. Therefore, the units place can be filed by any of the given $5$ digits.

Similarly, tens and hundreds digits can be filled in by any given $5$ digits.

Thus, by the multiplication principle, the number of ways in which three digit numbers can be formed from the given digits is $5×5×5=125$

$□□□HTO$

Now, there are $5$ ways to fill ones place.

Since, repetition is allowed , so tens place can also be filled by $5$ ways.

Similarly,hundreds place can also be filled by $5$ ways.

So, number of ways in which three digit numbers can be formed from the given digits is $5×5×5=125$

There will be as many way ways as there are ways of filling $3$ vacant places in succession by the given five digits. In this case, repetition of digits is allowed. Therefore, the units place can be filed by any of the given $5$ digits.

Similarly, tens and hundreds digits can be filled in by any given $5$ digits.

Thus, by the multiplication principle, the number of ways in which three digit numbers can be formed from the given digits is $5×5×5=125$

ii) A three digit number is to be formed from given $5$ digits $1,2,3,4,5$.

$□□□HTO$Now, there are $5$ ways to fill ones place.

Since, repetition is not allowed , so tens place can be filled by remaining $4$ digits.

So, tens place can be filled in $4$ ways.

Similarly, to fill hundreds place, we have $3$ digits remaining.

So,hundreds can be filled by $3$ ways.

So, required number of ways in which three digit numbers can be formed from the given digits is $5×4×3=60$