Question

How many numbers greater than 1000 can be formed from the digits 1,1,2,2,3?

Answer 1

Hint: First find the possible cases and then arrangements possible inside them. After finding the number of arrangements possible for all possible cases, look at definitions of product rule, sum rule and then apply the rule which is suitable at this position. After applying just simplify, basically to get the result you require arrangement of a things in which b are of same type, care of other same type is given by
$\dfrac{a!}{b!c!}$

Complete step-by-step answer:
Given conditions in the question, can be written as a number greater than 1000 formed by digits 1, 1, 2, 2, 3. So, the possible ways are the number being 4-digit, 5-digit.

Possible cases for 4 digits number:
Case-I digits taken are 1, 1, 2, 2
We know there are ‘a’ thing of ‘b’ are of same type we get arrangements = $\dfrac{a!}{b!}$
By using this here, we get the number of arrangements as:
$\dfrac{4!}{2!2!}............(i)$
Case-II Digits taken are 1, 1, 2, 3
We know if there are ‘a’ thing of which ‘b’ things are same,
number of arrangements = $\dfrac{a!}{b!}$
By using this, we get numbers of arrangement in this case are:
$\dfrac{4!}{2!}............(ii)$
Case- III digits taken all 2, 2, 3, 1
We know if there are ‘a’ thing of which ‘b’ things are same,
number of arrangements = $\dfrac{a!}{b!}$
By using this, we get numbers of arrangement in this case are:
$\dfrac{4!}{2!}............(iii)$
Case for a 5-digit number is: using the digits 1, 1, 2, 2, 3.
We know if there are ‘a’ thing of which ‘b’ things are same,
number of arrangements = $\dfrac{a!}{b!}$
By using this, we get numbers of arrangement in this case are:
$\dfrac{5!}{2!2!}............(iv)$
Rule of sum: In combination, the rule of sum or addition principle is basic counting principle. It is simply defined as, if there are A ways of doing P work and B ways of doing Q work. Given P, Q words cannot be done together. Total number of ways to do both P, Q are given by (A+B) ways.
Rule of product: In combinations the rule of product or multiplication principle is basic counting principle. It is simply defined as, if there are A ways of doing P work and B ways of doing Q work. Given P, Q works can be done at a time. Total number of ways to do both P, Q works are given by \[\left( A\times B \right)\]ways.
By looking at both definitions, we get we have to use sum rule:
Total= ways of $\left[ \left( 1,1,2,2 \right)+\left( 1,1,2,3 \right)+\left( 2,2,3,1 \right)+\left( 1,1,2,2,3 \right) \right]$
By substituting all values, we get the total as follows here:
Total $\dfrac{4!}{2!2!}+\dfrac{4!}{2!}+\dfrac{4!}{2!}+\dfrac{5!}{2!2!}=6+12+12+30$
Total = 60.
Therefore, 60 numbers greater than 1000 are possible.

Note: Be careful you must consider all cases if you miss even if you might lose the solution and lead to the wrong answer. While using formulas if you have more repetitions only denominator changes numerator always remains the same.