Question

How many different nine-digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions?
1). 16
2). 36
3). 60
4). 180

Answer 1

Hint: We have to arrange the digits in a way in which odd digits occupy the even positions. For that first we will find the possible ways in which the odd digits can be placed and the possible ways in which even digits can be placed. Then, we will multiply both the outcomes and we will get our answer.

Complete step-by-step solution:
According to the question,
We have to arrange the digits in a way that odd digits occupy the even places. We have 4 odd digits and 5 even digits. So, the arrangement can be:
EOEOEOEOE
(E = Even digit, O = Odd digit)
Arrangement of odd digits:
There are 4 even places and 4 odd digits which are 3355. So, we can arrange odd digits in \[\dfrac{{4!}}{{2!.2!}}\]ways. $\left( {\dfrac{{4!}}{{2!.2!}} = 6} \right)$
So, odd digits can be arranged in 6 ways.
Arrangement of even digits:
There are 5 odd places and 5 even digits which are 22888. So, we can arrange even digits in $\dfrac{{5!}}{{3!.2!}}$ ways. $\left( {\dfrac{{5!}}{{2!.3!}} = 10} \right)$
So, even digits can be arranged in 10 ways.
Total number of arrangements = $6 \times 10$
= $60$
We can arrange these nine-digits in 60 ways.
So, option (3) is the correct answer.

Note: The concept of permutation and combination are used in this kind of questions. It is must to create a sample arrangement by considering the conditions given in the questions. This helps in solving the question accurately.