Case study (4 Marks)

Question 14 Marks

Republic day is a national holiday of India. It honours the date on which the constitution of India came into effect on 26 January 1950 replacing the Government of India Act (1935) as the governing document of India and thus, turning the nation into a newly formed republic.

Answer the following question, which are based on the word "REPUBLIC".

(i) Find the number of arrangements of the letters of the word 'REPUBLIC'.
(a) 40300 (b) 30420 (c) 40320 (d) 40400

(ii) How many arrangements start with a vowel?
(a) 12015 (b) 15120 (c) 12018 (d) 15100

(iii) Which concept is used for finding the arrangements start with a vowel?
(a) Permutation (b) FPM (c) Combination (d) FPA

(iv) If the number of arrangements of the letters of the word 'REPUBLIC' is abcde, the (a + b + $\mathbf{c}+\mathbf{d}+\mathbf{e})$ is
(a) 10 (b) 9 (c) 8 (d) 15

(v) If the number of arrangements start with a vowel is abcde, then $(\mathbf{a}+\mathbf{b})-(\mathbf{d}+\mathbf{e})$ is
(a) 2 (b) 3 (c) 4 (d) 5

Answer

(i) (c) The letters in the word 'REPUBLIC' are all distinct. There are 8 letters in the given word. So, the number of arrangements are 8 ! i.e. 40320.

(ii) (b) The vowels in a given word are ' $\mathrm{E}, \mathrm{I}, \mathrm{U}$ '. If we start a word from vowel, we can choose 1 vowel from 3 vowels in ${ }^3 \mathbf{C}_1$ ways. Further, remaining 7 letters can be arranged in 7 ! ways.
$\therefore$ Total number of arrangements start with a vowel
$
={ }^3 \mathrm{C}_1 \times 7 !=3 \times 5040=15120
$

(iii) (c) Combination

(iv) (b) Since, number of arrangements are 40320 .
On comparing, we get
$
\begin{gathered}
a=4, b=0, c=3, d=2, e=0 \\
\text { So, } a+b+c+d+e=4+0+3+2+0=9
\end{gathered}
$

(v) (c) Since, number of arrangements are 15120
On comparing, we get
$
\begin{gathered}
a=1, b=5, c=1, d=2, e=0 \\
\therefore(a+b)-(d+e)=(1+5)-(2+0)=6-2=4
\end{gathered}
$

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Question 24 Marks

Five students Ajay, Shyam, Yojana, Rahul and Akansha are sitting in a playground in a line.

Based on the above information, answer the following questions.

(i) Total number of ways of sitting arrangement of five students is
(a) 120 (b) 60 (c) 24 (d) None of these

(ii) Total number of arrangement of sitting, if Ajay and Yojana sit together, is
(a) 60 (b) 48 (c) 72 (d) 120

(iii) Total number of arrangement 'Yojana and Rahul sitting at extreme position' is
(a) 24 (b) 36 (c) 48 (d) 12

(iv) Total number of arrangement, if shyam is sitting in the middle, is
(a) 24 (b) 12 (c) 6 (d) 36

(v) Total number of arrangement sitting Yojana and Rahul not sit together, is
(a) 72 (b) 120 (c) 60 (d) 144

Answer

(i) (a) We have five students.
$\therefore$ Total number of arrangements is $\mathbf{5} !=\mathbf{1 2 0}$

(ii) (b) Ajay and Yojana sit together.
Total number of arrangements $=4 ! \times 2 !=24 \times 2=48$

(iii) (d) Total number of arrangements of Yojana and Rahul sitting in extreme position is $2 ! \times 3 !=$ $2 \times 6=12$

(iv) (a) Number of arrangements of Shyam in middle is
$
4 !=24
$

(v) (a) Number of arrangement Yojana and Rahul not sit together, is
$
\frac{4 !}{2 !} \times 3 !=72 \text { or } 5 !-4 ! \times 2 !=72
$

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Maths Case study (4 Marks)

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