Question

The number ways of selecting five letters from the letters of the word $INDEPENDENT$ if

• A. atleast containing one pair of same letter is $66$
• B. without any ristriction is $72$
• C. $3$ letters are same is $24$
• D. atleast containing one pair of same letter is $42$

Answer 1

Answer: (A), (B), (C)

$3$ letters are same is $24$

Word $INDEPENDENT$
the letters are $3E,3N,2D,1I,1P,1T$
Now
Case $1:$ all the letters are distinct
${}^6{C}_5=6$
Case $2:$ $3$ letters are distinct and $2$ alike
${}^3{C}_1\times {}^5{C}_3=30$
Case $3:$ $2$ letters are distinct and $3$ alike
${}^2{C}_1\times {}^5{C}_2=20$
Case $4:$ $2$ alike, $2$ alike and $1$ distinct
​​​​​​​${}^3{C}_2\times {}^4{C}_1=12$
Case $4:$ $3$ alike and $2$ alike
${}^2{C}_1\times {}^2{C}_1=4$

Now number of ways selecting the five letters which contains atleast one pair of same letter $30+12+20+4=66$

Now number of ways selecting the five letters which contains $3$ same letter $20+4=24$

Total number of ways $=6+30+20+12+4=72$