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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Similar QuestionsQuestion 1: How many 6 digit numbers can be formed by using the digit 0,1,2,3,4,5. If repetition of digits is allowed? Answer:
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Without considering different cases. This is another way to get the same answer as already answerd above. Here we start by finding the total amount of the four-digit numbers with distinct digits, then finding the amount of odd digits, filling the same criteria, and last, we subtract the odd numbers from the total, to find the even numbers filling the criteria. We have totally seven digits. We know that the digit zero cannot be placed at the position representing the thousand position. That leaves us with six digits to choose from for this position and that can be made in $P(6,1) = \frac{6!}{(6-1)!} = \frac{6!}{5!}=6$, different ways. Now, we have three positions left to fill and six digits to choose from, including the digit zero, which can be placed anywhere in the remaining positions. This choice can be done in $P(6,3) = \frac{6!}{(6-3)!} = \frac{6!}{3!}=6 \cdot5 \cdot4$, different ways. Finally, with distinct digits, there is $6 \cdot6 \cdot5 \cdot4 =720$ four-digit numbers to be constructed. We know that the amount of odd numbers plus the amount of even numbers equal the total amount of the 720 four-digit numbers. The odd numbers are 1, 3 and 5. The question to be asked is how many of the 720 are odd? The digit to fill the unit position can only be chosen from the digits 1, 3 or 5, and this choice can be made in $P(3,1)=\frac{3!}{(3-1)!}=\frac{3!}{2!}=3$, different ways. To choose the digit filling the thousand position, we have five valid digits to choose from, since the zero digit is not valid for this position. The choice can be made in $P(5,1)=\frac{5!}{(5-1)!}=\frac{5!}{4!}=5$, different ways. Now we are left with two positions to fill, the hundred and tenth position, and five digits to choose from, now including the zero digit. The choice for these two positions can be made in $P(5,2)=\frac{5!}{(5-2)!}=\frac{5!}{3!}=5 \cdot4=20$, different ways. The total number of odd four-digit numbers, with distinct digits are $5 \cdot5 \cdot4 \cdot3 =300$. Now, we can answer the question how many of these 720, four-digit numbers, are even, by the subtraction $720-300=420$. The even numbers are 420. |