Question 6 and 7, Exercise 6.2

Solutions of Question 6 and 7 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

How many $4$-digit numbers can be formed with the digits $$1,2,3,4,5,6$ when the repetition of the digits is allowed?

Solution.

Possibilities to fill units place $6$
Possibilities to fill tens place $6$
Possibilities to fill hundreds place $6$
Possibilities to fill thousands place $6$
$$\text{Total possibilities }6 \times 6 \times 6 \times 6=6^4=1296$$

How many numbers can be formed with the digits $1,1,2,2,3,3,4$ so that the even digits always occupy the even places, using all the digits and no digit is repeated?

Solution.

Total number of digits given $=7$.
Even places are $2^{\text {nd }}$, fourth and $6^{\text {th }}$ place.
While odd places are $1^{\text {st }}, 3^{\text {rd }}, 5^{\text {th }}$ and $7^{\text {th }}$ place.
We have to arrange even numbers $2,2,4$ on even places and
odd values $1,1,3,3$ an odd places for even places.
$2$ is repeated twice so
Possible arrangements for even places $=\dfrac{3!}{2!}=\dfrac{6}{2}=3$
For odd place, $1$ is repeated twice and $3$ is repeated twice,
so possible arrangements for odd places $=\dfrac{4!}{2!2!}=\dfrac{24}{4}=6$

$$\text{Total possible numbers }=3 \times 6=18$$

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