Let N be the number of arrangements of the letters of the word "LOVEISLIFE" in which no two alike letters are together and containing the word LOVE. Then N is

$\text{LOVE}/\text{S}/\text{L}/\text{F}/\text{E}/\Rightarrow { }^6{\text{C}}_2\times {\left(1\right)}^2\times 5!$ 

(Fill I in the gaps)

Now $\text{LLOVE}/\text{S }/\text{ F }/\text{E }/\Rightarrow { }^5{\text{C}}_2\times {\left(1\right)}^2\times 4!$

Same for $\text{LOVEE}/\text{\hspace{0.17em}S}/\text{F}/\text{L}/\Rightarrow { }^5{\text{C}}_2\times {\left(1\right)}^2\times 4$

$\text{LLOVEE}/S/\text{F}/\Rightarrow { }^4{\text{C}}_2\times {\left(1\right)}^2\times 3!$

So Ans $={ }^6{\text{C}}_2\times {\left(1\right)}^2\times 5!-{ }^5{\text{C}}_2\times {\left(1\right)}^2\times 4!-{ }^5{\text{C}}_2\times {\left(1\right)}^2\times 4!+{ }^4{\text{C}}_2\times {\left(1\right)}^2\times 3!=1356$