Match the following

A). In the word ENGINEER the vowels are E,E,E,I, consonants are N,N,G,R Required = (No. of arrangements in which E,E,E,I are together) – (No. of arrangemnets in which EEE are together when E,E,E,I are together)

$=\left(\frac{\mathrm{\angle }5}{\mathrm{\angle }2}\times \frac{\mathrm{\angle }4}{\mathrm{\angle }3}\right)-\left(\frac{\mathrm{\angle }5}{\mathrm{\angle }2}\times 2\right)=60\times (4-2)=120=5!$

B). No.of arrangements of E,E,E and Consonants L,V,N alternately starting with E is $\mathrm{\angle }3$ and starting with a
consonant is  $\mathrm{\angle }3$ K=Total no.of such arrangments = $\mathrm{\angle }3+\mathrm{\angle }3=12$ 

$\Rightarrow 10K=120=5\text{ ! }$

C). P = No.of Derangements of 5 letters in 5 addresed envelops $\Rightarrow P=D_5=44$ 

44 is divisible by 22 where $\frac{1}{3}\left({ }^{12}{C}_2\right)=22$

D). Using 1,2,2,3,3,3 numbers of 6 digits numbers that are starting with 123, 132,133 or 2 are greater than 123000. No.of such numbers =$\frac{\mathrm{\angle }3}{\mathrm{\angle }2}+\frac{\mathrm{\angle }3}{\mathrm{\angle }2}+\frac{\mathrm{\angle }3}{\mathrm{\angle }2}+\frac{\mathrm{\angle }5}{\mathrm{\angle }3}+\frac{\mathrm{\angle }5}{\mathrm{\angle }2\cdot \mathrm{\angle }2}=59$