Question

The number of ways in which the time table for Monday can be made if there must be $5$ lessons to be taught that day (Algebra, Geometry, Calculus, Trigonometry, Vector) and topics Algebra and Geometry cannot immediately follow each other are

• A. $72$
• B. $5!$
• C. $3\times \frac{5!}{2}$
• D. $6!$

Answer 1

The correct option is A $72$

$\times V\times C\times T\times$
On these 4 crosses we can arrange two subjects (Algebra and Geometry) in ${}^4{P}_2$.
And $V,C,T$ can be arranged in $3!$ ways.
Hence, total number of ways ${=}^4{P}_2.3!=72$

Alternate :
The number of required ways $=$ Total ways without restriction $-$ Total ways when Algebra and Geometry are one after the other.
Total ways without restriction $=5!$
Total ways when Algebra and Geometry are one after the other $=2!\times 4!$
$($Consider Algebra and Geometry as one object and interchange among themself$)$
The number of required ways $=5!-2!\times 4!=72$