Question

Fill in the blanks.
(i) The number of different words that can be formed from the letters of the word INTERMEDIATE such that two vowels never come together is P .
(ii) Three balls are drawn from a bag containing $5$ red, $4$ white and $3$ black balls. The number of ways in which this can be done if atleast $2$ are red is Q.
(iii) The total number of ways in which six ${}^{\prime }{+}^{\prime }$ and four ${}^{\prime }{-}^{\prime }$ signs can be arranged in a line such that no two signs ${}^{\prime }{-}^{\prime }$ occur together, is R .

	P	Q	R
(a)	160002	90	35
(b)	151200	80	35
(c)	162200	85	40
(d)	153400	105	40

• A. a
• B. b
• C. c
• D. d

Answer 1

Answer: (B)

Total number of letters in the word ‘INTERMEDIATE’ $=12$
Number of consonants $=6$,
Number of vowels $=6$
When we fix consonants at six places then there are seven places for vowels as shown below.

$6$ consonants out of which $2$ are alike can be placed in $\frac{6!}{2!}$
ways and $6$ vowels, out of which $3E’s$ alike and $2I^{\prime }s$ can be arranged at seven places in
${}^7{P}_6\times \frac{1}{3!}\times \frac{1}{2!}$ ways.
$\therefore$ Total number of words
$=\frac{6!}{2!}\times {}^7{P}_6\times \frac{1}{3!}\times \frac{1}{2!}=151200$
(ii) Required number of ways
$=\left({}^5{C}_2\times {}^7{C}_1\right)+{}^5{C}_3=\left(10\times 7\right)+10$
$=70+10$
$=80$
(iii) The arrangement of signs is shown in the following figure.

Thus, ${}^{\prime }{+}^{\prime }$ sign can be arranged in $1$ way because all are identical and $4$ negative signs can be arranged at $7$ places . in ${}^7{C}_4$ ways.
$\therefore$ Total number of ways $={}^7{C}_4\times 1=35$