Answer VerifiedHint: In the above question we need to find the number of ways to form $4$-digit numbers. If all four digits are different, we use $^5{P_4}$. If there is one repeated digit, there are $5$ ways to choose which digit is repeated. Then there are $^4{C_2} = 6$ ways to place the repeated digit and $^4{P_2}$ = 12 ways to place the non – repeated digits. And similarly, we arrange numbers when there are two repeated digits. Complete step by step solution: According to the question, we have to from $4$-digit number from digits $1, 1, 2, 2, 3, 3, 4, 4, 5, 5$.We can form $4$-digit number in three ways –When all the four digits are different, then the number of 4-digit number can be formed is = $^5{P_4} $$= \dfrac{{5!}}{{1!}} $$= \dfrac{{5 \times 4 \times 3 \times 2 \times 1}}{1} $$= 120$ways.When two digits are same and another two digits are different, then the number of 4-digit numbers can be formed is =$^5{C_1}{ \times ^4}{C_2}{ \times ^4}{P_2} $$= \dfrac{{5!}}{{4! \times 1!}} \times \dfrac{{4!}}{{2! \times 2!}} \times \dfrac{{4!}}{{2!}}$$ = \dfrac{{5 \times 4!}}{{4! \times 1}} \times \dfrac{{4 \times 3 \times 2!}}{{2! \times 2 \times 1}} \times \dfrac{{4 \times 3 \times 2!}}{{2!}} $$= 5 \times 6 \times 12 $$= 360$ waysWhen there are two repeated digits, then the number of 4-digit numbers can be formed is = $^5{C_2}$$ = \dfrac{{5!}}{{3! \times 2!}}$$ = \dfrac{{5 \times 4 \times 3!}}{{3! \times 2 \times 1}} $$= 10$There are $6$ ways to place the first repeated digit and $1$ way to place the second repeated digit.Thus, we have; Total number of ways when there are two repeated digits are$ = 10 \times 6 \times 1 = 60 $waysTotal number of ways to form 4-digit numbers from given digits are$ = 120 + 360 + 60$$ = 580 $ ways.$\therefore $Total number of ways to form 4-digit numbers from given digits = 580. Note: In these types of questions use the permutation concept. Permutations are for lists where order matters and combinations are for groups where order doesn’t matter.In mathematics, permutation relates to the function of ordering all the members of a group into some series or arrangement. In other words, if the group is already directed, then the redirecting of its components is called the process of permuting. Permutations take place, in more or less important ways, in almost every district of mathematics. They frequently appear when different commands on certain limited places are observed. Permutation A permutation is known as the process of organizing the group, body, or numbers in order, selecting the or numbers from the set, is known as combinations in such a way that the sequence of the integer does not bother. Permutation Formula In permutation, r items are collected from a set of n items without any replacement. In this sequence of collecting matter.
Combination The combination is a way of choosing objects from a group, such that (unlike permutations) the sequence of choosing does not matter. In smaller cases, it is imaginable, to sum up, the number of combinations. Combination refers to the combination of n objects taken k at a time without repetition. To mention combinations in which repetition is allowed, the expressions k-selection or k-combination with repetition are frequently used. Combination Formula In combination, r objects are selected from a group of n objects and where the sequence of selecting does not matter.
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