Question

All possible numbers are formed using the digits $1, \, 1, \, 2, \, 2, \, 2, \, 2, \, 3, \, 4, \, 4 \, $ taken all at a time. The number of such numbers in which the odd digits occupy even places is

• A. $175$
• B. $162$
• C. $180$
• D. $160$

Answer 1

Answer: (C)

Odd digits: $1, \, 1, \, 3$
Even digits: $2, \, 2, \, 2, \, 2, \, 4, \, 4$
The number of ways of placing odd digits at even position
$={}^{4}C_{3}^{}\times \frac{3 !}{2 !}=4\times 3=12$
The number of ways of placing even digits $=\frac{6 !}{4 ! \, 2 !}=15 \, $
$\therefore $ Total number of ways $=12\times 15=180$