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Permutations — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsPermutations2 Marks
Question
How many different five$-$digit number licence plates can be made if.
  1. First digit cannot be zero and the repetition of digits is not allowed.
  2. The first$-$digit cannot be zero, but the repetition of digits is allowed?
✓

Answer

  1. Zero cannot be first digit of the license plates.
This means the first digit can be selected from the $9$ digits $1, 2, 3, 4 .... , 9$
So, there are $9$ ways of filling the first digit of the license plates.
Now, $9$ digits are Ieft including $0.$
So, second place can be filled with any of the remaining $9$ digits in $9$ ways.
The third place of the license plates can be filled with in any of the remaining $8$ digits.
So, there are $8$ ways of filling the third place.
The fourth place of the license plates can be filled with in any of the remaining $7$ digits.
So, there are $7$ ways at filling the fourth place.
The last place of the license plates can be filled with in any of the remaining $6$ digits.
So, there are $6$ ways of filling the fourth place.
Hence, the total number of ways $= 9 \times 9 \times 8 \times 7 \times 6 = 27216$
  1. Zero cannot be first digit of the license plates.
$\therefore$ First digit can be selected from the $9$ digits $1, 2, 3 ..... , 9$
​​​​​​​So, there are $9$ ways at filling the first digit of the licence plates.
The repetition of digits is allowed to made a license plates number.
$\therefore$ The number of ways to fill the remaining places of the number platas $= 10 \times 10 \times 10 \times 10.$
Hence, the total number of ways $= 9 \times 10 \times 10 \times 10 \times 10 = 90,000.$

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