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A Five digits number is written down at random. Find the probability that number is divisible by 5 and no two consecutive digits are identical.
Asked by rairp2017 | 27 May, 2021, 12:21: AM
Let us find out  first , how many ways a 5 digit number is written

Let us consider position number of left most digit is 1, right most digit is 5 and position number of
in-between digita are assigned as shown in above figure.

Number of ways of writing digits at position number 1 is 9 because digit zero can not be chosen out of 10 digits .

Number of ways of writing digits at position number 2 is 9 because one digit is already selected for position number 1.

Number of ways of writing digits at position number 3 is 8 because two digits are already selected for previous positions.

Number of ways of writing digits at position number 4 is 7 because three digits are already selected for previous positions.

Number of ways of writing digits at position number 5 is 6 because four digits are already selected for previous positions.

Hence number of ways writing the 5 digit number , N = ( 9 × 9 × 8 × 7 × 6 )

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Let us find out  now , how many ways a 5 digit number that is divisible by 5 is written

To get a 5-digit number that is divisible by 5 , digit in units place hase to be either 5 or 0 .

Number of ways of writing digits at position number 1 is 9 because digit zero can not be chosen out of 10 digits .

Number of ways of writing digits at position number 2 is 8 because one digit is already selected for position number 1
and either 0 or 5 is reserved for position number 5.

Number of ways of writing digits at position number 3 is 7 because two digits are already selected for previous positions
and either 0 or 5 is reserved for position number 5.

Number of ways of writing digits at position number 4 is 6 because thre digits are already selected for previous positions
and either 0 or 5 is reserved for position number 5.

Number of ways of writing digits at position number 5 is 2 only

Hence number of ways writing the 5 digit number that is divisble by 5, n = ( 9 × 8 × 7 × 6 × 2 )

Probability of writing a 5 digit number that is divisible by 5 , P = n/N

P = ( 9 × 8 × 7 × 6 × 2 ) / ( 9 × 9 × 8 × 7 × 6 )  = 2 / 9
Answered by Thiyagarajan K | 20 Sep, 2021, 11:53: PM

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