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 ______.

Options

• 180

• 175

• 162

• 160

Answer 1

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 180.

Explanation:

1, 1, 2, 2, 2, 2, 3, 4, 4,

Odd digit in even place

4 Even places and 2, 2, 2, 2, 4, 4 6 numbers which are even so 1, 1, 3 only 3 odd places that can be placed in 4 Even places ${\displaystyle {}^4{\text{C}}_3\times \frac{3!}{2!}}$ for even place.

Remaining Even numbers in 6 places by ${\displaystyle \frac{6!}{4!2!}}$

So total ways ${\displaystyle \times \frac{3!}{2!}\times \frac{61}{4!2!}}$ = 180