In how many ways $4$ letters can be selected from the letters of the word INEFFECTIVE?

INEFFECTIVE contains $11$ letters: $EEE, FF, II, C, T, N, V$. We will make following cases to select $4$letters.

Case 1: 3 alike and 1 distinct

$3$ alike letters can be selected from $1$ set of $3$ alike letters $\left(EEE\right)$ in 1 way.

⇒ The number of ways to select $3$ alike letters $=1$

⇒ The number of ways to select $1$ distinct letters $=6$

⇒ Total ways $=6\times 1=6$

Case 2: 2 alike and 2 alike

$\text{'}2$alike and $2$ alike' means we have to select $2$ groups of $2$ alike letters $(EE, FE, II)$ in  ${}^3{C}_2$ ways.

$\Rightarrow$The number of ways to select $\text{'}2$ alike and $2$ alike' letters $={}^3{C}_2=3$

Case 3: 2 alike and 2 distinct

$1$ group of $2$ alike letters can be selected from $3$ sets of $2$ alike letters $(EE, FE, II)$ is  ${}^3{C}_1$ ways.

$2$ distinct letters can be selected from $6$ distinct letters ($\mathrm{C},\mathrm{T},\mathrm{N},\mathrm{V},$ remaining $2$ sets of two letters alike) in ${}^6{C}_2$ ways.

The number of ways to select $\text{'}2$ alike and $2$ distinct letters'

${}^3{C}_1\times {}^6{C}_2=3\times 15=45$

Case 4: AII distinct letters

All distinct letters can be selected from $7$ distinct letters $(I, E, F, N, C, T, V)$ in ${}^7{C}_4$ ways.

$\Rightarrow$The number of ways to select all distinct letters $={}^7{C}_4=35$

Combining (1), (2), (3) and (4), we get,

Total number of ways to select $4$ letters from the letter word 'INEFFECTIVE' $=6+3+45+35=89$