Question

An old man while dialing a $7$ digit telephone number remembers that the first four digits consists of one $1^{\prime }s$, one $2^{\prime }s$ and two $3^{\prime }s$. He also remembers that the fifth digit is either a $4$ or $5$ while has no memorizing of the sixth digit, he remembers that the seventh digit is $9$ minus the sixth digit. Maximum number if distinct trails he has to try make sure that he dials the correct telephone number, is

• A. $360$
• B. $240$
• C. $216$
• D. $none$

Answer 1

The correct option is B $240$

Number of telephone digits $=7$

There are one 1's, one 2's and two 3's

Fifth Digits $=4or5$

Possibilities of 6th and 7th digits$=(0,9),(1,8),(7,2),(6,3),(5,4),(4,5),(3,6),(2,7),(1,8),(9,0)$

Number of possibilities for first $4$ digits

$=\frac{4!}{2!}$ ways

There are two possibilities for $5$th digits.

$6$th digit has $10$ possibilities.

So, the maximum number of trails to get the correct telephone number$=\frac{4!}{2}\times 2\times 10$

$=240$