How many different five-digit number license plates can be made if :i.

1.	How many different five-digit number license plates can be made if :i. The first digit cannot be zero, and the repetition of digits is not allowed, ii. The first-digit cannot be zero, but the repetition of digits is allowed?
Answer» (i) Given : Five-digit number is required in which the first digit cannot be zero, and the repetition of digits is not allowed. Assume five boxes, now the first box can be filled with one of the nine available digits, so the possibility is ⁹C₁ Similarly, The second box can be filled with one of the nine available digits, so the possibility is ⁹C₁ The third box can be filled with one of the eight available digits, so the possibility is ⁸C₁ The fourth box can be filled with one of the seven available digits, so the possibility is ⁷C₁ The fifth box can be filled with one of the six available digits, so the possibility is 6C1 Hence, The number of total possible outcomes is ⁹C₁ × ₉C¹ × ⁸C₁ × ⁷C₁ × ⁶C₁ = 9 × 9 × 8 × 7 × 6 = 27216 (ii) Given : Five-digit number is required in which the first digit cannot be zero, and the repetition of digits is allowed Assume five boxes, now the first box can be filled with one of the nine available digits, so the possibility is ⁹C₁ Similarly, The second box can be filled with one of the ten available digits, so the possibility is ¹⁰C₁ The third box can be filled with one of the ten available digits, so the possibility is ¹⁰C₁ The fourth box can be filled with one of the ten available digits, so the possibility is ¹⁰C₁ The fifth box can be filled with one of the ten available digits, so the possibility is ¹⁰C₁ Hence, The number of total possible outcomes is ⁹C₁ × ¹⁰C₁ × ¹⁰C₁ × ¹⁰C₁ × ¹⁰C₁ = 9 × 10 × 10 × 10 × 10 = 90000

How many different five-digit number license plates can be made if :i. The first digit cannot be zero, and the repetition of digits is not allowed, ii. The first-digit cannot be zero, but the repetition of digits is allowed?

Answer»

(i) Given : Five-digit number is required in which the first digit cannot be zero, and the repetition of digits is not allowed.

Assume five boxes, now the first box can be filled with one of the nine available digits, so the possibility is ⁹C₁

Similarly,

The second box can be filled with one of the nine available digits, so the possibility is ⁹C₁

The third box can be filled with one of the eight available digits, so the possibility is ⁸C₁

The fourth box can be filled with one of the seven available digits, so the possibility is ⁷C₁

The fifth box can be filled with one of the six available digits, so the possibility is 6C1

Hence,

The number of total possible outcomes is ⁹C₁ × ₉C¹ × ⁸C₁ × ⁷C₁ × ⁶C₁

= 9 × 9 × 8 × 7 × 6

= 27216

(ii) Given : Five-digit number is required in which the first digit cannot be zero, and the repetition of digits is allowed

Assume five boxes, now the first box can be filled with one of the nine available digits, so the possibility is ⁹C₁

Similarly,

The second box can be filled with one of the ten available digits, so the possibility is ¹⁰C₁

The third box can be filled with one of the ten available digits, so the possibility is ¹⁰C₁

The fourth box can be filled with one of the ten available digits, so the possibility is ¹⁰C₁

The fifth box can be filled with one of the ten available digits, so the possibility is ¹⁰C₁

Hence,

The number of total possible outcomes is ⁹C₁ × ¹⁰C₁ × ¹⁰C₁ × ¹⁰C₁ × ¹⁰C₁

How many different five-digit number license plates can be made if :i. The first digit cannot be zero, and the repetition of digits is not allowed, ii. The first-digit cannot be zero, but the repetition of digits is allowed?

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