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

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

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/m-fsperh91g4r8tyqf.jpg" /> Odd digits: 1, 1, 3 Even digits: 2, 2, 2, 2, 4, 4 The number of ways of placing odd digits at even position =4C3× (3 !/2 !)=4× 3=12 The number of ways of placing even digits =(6 !/4 ! 2 !)=15 ∴ Total number of ways =12× 15=180

Q. 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

$175$

$162$

$180$

$160$

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 $= \frac{6 !}{4 ! 2 !} = 15$
$∴$ Total number of ways $= 12 \times 15 = 180$

Q. 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

175

162

180

160