In how many ways letters of word lovely can be arranged

The word ARRANGEMENT has $11$ letters, not all of them distinct. Imagine that they are written on little Scrabble squares. And suppose we have $11$ consecutive slots into which to put these squares.

There are $\dbinom{11}{2}$ ways to choose the slots where the two A's will go. For each of these ways, there are $\dbinom{9}{2}$ ways to decide where the two R's will go. For every decision about the A's and R's, there are $\dbinom{7}{2}$ ways to decide where the N's will go. Similarly, there are now $\dbinom{5}{2}$ ways to decide where the E's will go. That leaves $3$ gaps, and $3$ singleton letters, which can be arranged in $3!$ ways, for a total of $$\binom{11}{2}\binom{9}{2}\binom{7}{2}\binom{5}{2}3!.$$

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1. How many ways can the letters in the word LOVE be arranged?

Answer:

6.12x11x10x9x8x7x6x5x4x3x2x1= 479,001,600 ways

5.?

8?

9.if no repetition 6x5x4x3x2x1 = 2,580

if pwede repetition 9x8x7x6x5z4x3x2x1= 362,880

Step-by-step explanation:

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The 4 letters word LOVE can be arranged in 24 distinct ways. The below detailed information shows how to find how many ways are there to order the letters LOVE and how it is being calculated in the real world problems.

Distinguishable Ways to Arrange the Word LOVE
The below step by step work generated by the word permutations calculator shows how to find how many different ways can the letters of the word LOVE be arranged.

Objective: Find how many distinguishable ways are there to order the letters in the word LOVE.

Step by step workout:

Input parameters and values:

n1(L) = 1, n2(O) = 1, n3(V) = 1, n4(E) = 1

step 2 Apply the values extracted from the word LOVE in the (nPr) permutations equation

= 1 x 2 x 3 x 4/{(1) (1) (1) (1)}

= 24/1

Apart from the word LOVE, you may try different words with various lengths with or without repetition of letters to observe how it affects the nPr word permutation calculation to find how many ways the letters in the given word can be arranged.