2. Aim: To find the no. of ways can four of the letters E, N, G, I, in the given
word is arranged.
Solution:
S1)
Letter Sequence f(x)
E 0,1,2 1+
𝑥
1!
+
𝑥2
2!
N 0,1,2 1+
𝑥
1!
+
𝑥2
2!
G 0,1 1+
𝑥
1!
I 0,1 1+
𝑥
1!
S2) E(x) =(1+
𝑥
1!
+
𝑥2
2!
) (1+
𝑥
1!
+
𝑥2
2!
)( 1+
𝑥
1!
)( 1+
𝑥
1!
)
S3) Method I
The sequence ans=co.eff of
𝑥4
4!
in E(x)
E(x) =(1+
𝑥
1!
+
𝑥2
2!
𝑥 +
𝑥2
2!
+
𝑥3
2!
+
𝑥2
2!
+
𝑥3
2!
+
𝑥4
2!2!
)
E(x) =(1+2x+3𝑥2
+𝑥3
+
𝑥4
4
)(1+2x+2𝑥2
)
E(x) =Co.eff of x4 [2+2+
1
4
]
=co.eff of x4[4+
1
4
]
E(x) =co.eff of
𝑥4
4!
[(
17
4
)4!]
∴ The sequence ans=co.eff of
𝑥4
4!
In E(x)
The sequence ans=
17
4
. 4!=
17
4
× 4 × 3 × 2 × 1=102
S4)Method II P=
𝑚!
𝑛1!𝑛2!𝑛3!…
3. Possible selection of four letters N0. Of Arrangement
E E G N 4!
2!1!1!…
=12
E E N N 4!
2!2!…
=6
E E I N 4!
2!1!1!…
=12
E E G I 4!
2!1!1!…
=12
E G N N 4!
2!1!1!…
=12
E I N N 4!
2!1!1!…
=12
G I N N 4!
2!1!1!…
=12
E I G N 4!
2!1!1!…
=24
102
Hence 102 ways 4 letters word can be arranged from the given word ENGINE
5) A ship carries 48 flags, 12 each of the colours Red, White, Blue, and Black.
Twelve of these flags are placed on a vertical pole in order to communicate a
signal to other ships.
a) How many of these signals use an even number of blue flags and an odd
number of black flags?
b) How many of the signals have atleast three white flags or no white flags or
no white flags at all?
Given:
Total no. of flags =48
No. of colours=4
No. of flags in a signal =12
Aim: To find no. of ways the 12 signal flags arrange such that
i) Even no. of blue and off no. of blacks
ii) Atleast 3 white or no white flag
Solution:
i)