Permutations and Combinations 1 Question 12

12. Consider all possible permutations of the letters of the word ENDEANOEL.

$(2008,6 M)$

Column I Column II
A. The number of permutations containing the word
ENDEA, is
p. $5 !$
B. The number of permutations in which the letter E
occurs in the first and the last positions, is
q. $2 \times 5 !$
C. The number of permutations in which none of the
letters D, L, N occurs in the last five positions, is
r. $7 \times 5 !$
D. The number of permutations in which the letters A, E,
O occur only in odd positions, is
s. $21 \times 5 !$
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Answer:

Correct Answer: 12. ($(A \rightarrow p ; B \rightarrow s ; C \rightarrow q ; D \rightarrow q)$)

Solution:

  1. A. If ENDEA is fixed word, then assume this as a single letter. Total number of letters $=5$

Total number of arrangements $=5$ !.

B. If $E$ is at first and last places, then total number of permutations $=7 ! / 2 !=21 \times 5$ !

C. If $D, L, N$ are not in last five positions

$\leftarrow D, L, N, N \rightarrow \leftarrow E, E, E, A, O \rightarrow$

Total number of permutations $=\frac{4 !}{2 !} \times \frac{5 !}{3 !}=2 \times 5$ !

D. Total number of odd positions $=5$

Permutations of AEEEO are $\frac{5 !}{3 !}$.

Total number of even positions $=4$

$\therefore$ Number of permutations of N, N, D, L $=\frac{4 !}{2 !}$

$\Rightarrow$ Total number of permutations $=\frac{5 !}{3 !} \times \frac{4 !}{2 !}=2 \times 5$ !



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