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Byju's Answer

Standard XII

Mathematics

Properties of Set Operation

Shubham has t...

Question

Shubham has to make a telephone call to his friend Nisheeth ,Unfortumately he does not remember the 7 digit phone number . But he remembers that the first three digits are 635 or 674 , the number is odd and three is exactly one 9 in the number .The maximum number of trials that Shubham has to make to be successful is a four digit number of the form abcd then c equal.

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Solution

Consider the problem

Let us assume that he starts out with the initial digits as $635$ .
In this case,
He would have to work on the following combination presuming that he places the number first time in fourth digit and next time in the fifth and then in the sixth and then finally in the seventh

first three digit fourth digit fifth digit sixth digit seventh digit total number of trials
$635$ $1 (N o . 9)$ $9 (0 t o 8)$ $9 (0 t o 8)$ $4 (1, 3, 5, 7)$ $9 \times 9 \times 4 = 324$
$635$ $9 (0 t o 8)$ $1 (N o . 9)$ $9 (0 t o 8)$ $4 (1, 3, 5, 7)$ $9 \times 9 \times 4 = 324$
$635$ $9 (0 t o 8)$ $9 (0 t o 8)$ $1 (N o . 9)$ $4 (1, 3, 5, 7)$ $9 \times 9 \times 4 = 324$
$635$ $9 (0 t o 8)$ $9 (0 t o 8)$ $9 (0 t o 8)$ $1 (N o . 9)$ $9 \times 9 \times 9 = 3729$

The total number of combinations with $635$ as prefix are

$324 + 324 + 324 + 729 = 1701$
He will have to repeat these $1701$ combinations with $674$ as prefix.

So in all, he will have to be prepared to run through $1701 + 1701 = 3402$
Before he can be certain to succeed.

Hence,

$c = 0$

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first three digit	fourth digit	fifth digit	sixth digit	seventh digit	total number of trials
$635$	$1 (N o . 9)$	$9 (0 t o 8)$	$9 (0 t o 8)$	$4 (1, 3, 5, 7)$	$9 \times 9 \times 4 = 324$
$635$	$9 (0 t o 8)$	$1 (N o . 9)$	$9 (0 t o 8)$	$4 (1, 3, 5, 7)$	$9 \times 9 \times 4 = 324$
$635$	$9 (0 t o 8)$	$9 (0 t o 8)$	$1 (N o . 9)$	$4 (1, 3, 5, 7)$	$9 \times 9 \times 4 = 324$
$635$	$9 (0 t o 8)$	$9 (0 t o 8)$	$9 (0 t o 8)$	$1 (N o . 9)$	$9 \times 9 \times 9 = 3729$