Question

How many five-digit number license plates can be made if

(i) 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 1

i) first digit cannot be zero and the repetition of digits is not allowed.

Let us assume 5 boxes shown above ,

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

$^{9}C_{1} = 9$

$Second \:box\:can \:be \:filled \:with \:one \\of \:the \:9 \: available \:digits ,\:so \:the \: possibility \\is \: ^{9}C_{1} = 9$

$Third \:box\:can \:be \:filled \:with \:one \\of \:the \:8 \: available \:digits ,\:so \:the \: possibility \\is \: ^{8}C_{1} = 8$

$Fourth\:box\:can \:be \:filled \:with \:one \\of \:the \:7\: available \:digits ,\:so \:the \: possibility \\is \: ^{7}C_{1} = 7$

$Fifth\:box\:can \:be \:filled \:with \:one \\of \:the \:6 \: available \:digits ,\:so \:the \: possibility \\is \: ^{6}C_{1} = 6$

$\red { Number \:of \: total \: possible }\\\red{outcomes \:is } = 9 \times 9 \:times 8 \times 7\times 6 \\= 27216$

(ii) The first digit cannot be zero, but the repetition of digits is allowed

$Let \:us \: assume \:5 \:boxes \:,now \:the \\first \:box \:can \:be \:filled \:with \:one \:of \: the \\9\: available \:digits , \\so \: possibility \: ^{9}C_{1} = 9$

$The \:second \: box \:can \:be \: filled \:with \\one \: of \:the \: 10 \: available \:digits , \\so \:the \: possibility \:is \: ^{10}C_{1} = 10$

$The \:third\: box \:can \:be \: filled \:with \\one \: of \:the \: 10 \: available \:digits , \\so \:the \: possibility \:is \: ^{10}C_{1} = 10$

$The \: Fourth \: box \:can \:be \: filled \:with \\one \: of \:the \: 10 \: available \:digits , \\so \:the \: possibility \:is \: ^{10}C_{1} = 10$

$The \:Fifth \: box \:can \:be \: filled \:with \\one \: of \:the \: 10 \: available \:digits , \\so \:the \: possibility \:is \: ^{10}C_{1} = 10$

$\red { Number \:of }\\\red{ total \: possible \: outcomes } = 9 \times 10 \times 10 \times 10 \:times 10 = 90000$

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