Factorials Expansion Finding Values multiplication division addition subtraction

No. Problems & Solutions [Click Hide/Show to display the solutions below the question]
01.
Is 3! + 4! = 7!

Solution » Hide/Show

3! + 4!=3! + 4 × 3!
 =3! (1 + 4)
 =(3 × 2) × (5)
 =6 × 5
 =30
7!=7 × 6 × 5 × 4 × 3 × 2 × 1
 =5040
∴ 3! + 4!7!
02.
Is (3!) (4!) = 12! ?

Solution » Hide/Show

3! × 4!=(3 × 2 × 1) (4 × 3 × 2 × 1)
 =6 × 24
 =144
12!=12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
 =47,90,01,600
∴ 3! × 4!12!
03.
Is
8!
4!
= 2!

Solution » Hide/Show

L.H.S =
8!
4!
  =
8 × 7 × 6 × 5 × 4!
4!
 =8 × 7 × 6 × 5
 =1680.
R.H.S=2!
 =2 × 1
 =2
⇒ L.H.SR.H.S
8!
4!
  2!
04.
Evaluate
(8!)2
(6!)3

Solution » Hide/Show

(8)!2
(6!)3
=
(8)! (8)!
(6)! × (6)! × (6)!
 =
8 × 7 × 6! × 8 × 7 × 6!
6! × 6! × 6 × 5 × 4 × 3 × 2 × 1
 =
8 × 7 × 8 × 7
6 × 5 × 4 × 3 × 2
 =
196
45
05.
Find the value of
(n − r + 1)!
(n − r − 1)!

Solution » Hide/Show

(n − r + 1)!
(n − r − 1)!
=
(n − r + 1) (n − r) (n − r − 1)!
(n − r − 1)!
  = (n − r + 1) (n − r)
  = n2 − n r − r n + r2 + n − r
  = n2 − 2 r n + n + r2− r
06.
Show that
(n + 3)!
n!
= n3+6 n 2 + 11 n + 6

Solution » Hide/Show

L.H.S =
(n + 3)!
n!
 =
(n + 3) [(n + 3) − 1] [(n + 3) − 2] [(n + 3 ) − 3]!
n!
 =
(n + 3) (n + 2) (n + 1) (n)!
n!
 = (n + 3) (n + 2) (n + 1)
 = (n2 + 3 n + 2 n + 6) (n + 1)
 = (n2 + 5 n + 6) (n + 1)
 = (n3 + n2 + 5 n2 + 5 n + 6 n + 6)
 = (n3 + 6n 2 + 11 n + 6)
 = R.H.S
07.
Prove that n! (n + 2) = n! + (n + 1)!

Solution » Hide/Show

R.H.S = n! + (n + 1)!
  = n! + (n + 1) (n)!
  = n! [1 + (n + 1)]
  = n! [n + 2]
  = L.H.S
08.
Prove that ( n + 1)! − n! = n.n!

Solution » Hide/Show

(n + 1)! − n! = (n + 1) n! − n!
  = n! [n + 1 − 1]
  = n! [n]
  = n. n!
  = R.H.S
09.
Show that
1
8!
+
1
9!
+
1
10!
=
101
10!

Solution » Hide/Show

L.H.S =
1
8!
+
1
9!
+
1
10!
        =
9 × 10 + 10 + 1
10!
  [LCM of 8!, 9!, 10! is 10!]
  =
90 + 10 + 1
10!
  =
101
10!
  = R.H.S
10.
1
n!
+
1
(n + 1)!
+
1
(n + 2)!
= ?

Solution » Hide/Show

 
11.
Prove that 19.18.......6.5 =
19!
4!

Solution » Hide/Show

R.H.S =
19!
4!
  =
19 × 18 × ...... 6 × 5 × 4!
4!
 =19 × 18 × ....... 6 × 5
 =L.H.S
Alt  
LHS: = 19 × 18 × --------- 6 × 5 × 1
 
=
19 × 18 × --------- 6 × 5 ×
4!
4!
  =
19 × 18 × ..... 6 × 5 × 4!
4!
  =
19!
4!
  = R.H.S
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No. Problems for Practice
01.
A. Find 5! B. Find 4! C. Find 6! D. Compute 3! + 4!
02.
A. Compute (5!) (3!)
03.
A.
Find
9!
6!
B.
Find the value of
10!
4!
C.
Evaluate
12!
8!
D.
What is the value of
8!
4!
?

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