Three digit numbers xyz are formed with digits 0- 1- 2- - 9 such that x - y - z then number of such numbers is

X lt; y lt; z → ^9C_3 x = y lt; z → ^9C_2 x lt; y = z → ^9C_2 x = y = z → ^9C_1 nbsp; nbsp; nbsp; nbsp; nbsp; nbsp; nbsp; nb ...

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Standard XIII

Mathematics

Combination with Restrictions

Three digit n...

Question

Three digit numbers $x y z$ are formed with digits $0, 1, 2, \dots 9$ such that $x \leq y \leq z$ then number of such numbers is

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Solution

x < y < z $\to^{9} C_{3}$
x = y < z $\to^{9} C_{2}$
x < y = z $\to^{9} C_{2}$
x = y = z $\to^{9} C_{1}$
_______
Total numbers $=^{9} C_{3} +^{9} C_{2} +^{9} C_{2} +^{9} C_{1} = 165$

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Similar questions

Q. Three digit numbers

x y z

are formed with digits

0, 1, 2, \dots 9

such that

x \leq y \leq z

then number of such numbers is

Find the total number of ways in which a 3- digit number xyz can be formed from the digits $1, 4, 5, 6, 7, 8 a n d 9$ such x,y,z that are in A.P and $x \neq y \neq z .$

Q. The number of three digit numbers of the form xyz such that x < y and

z \leq y

is
If zero is included it will be at

^{9} C_{2}

no. s
If zero is excluded \( \left\{

\begin{matrix} x, y, z a l l d i f f . & \Rightarrow & ^{9} C_{3} \times 2! x = z < y & \Rightarrow & ^{9} C_{2} x < y = z & \Rightarrow & ^{9} C_{2} N o^{'} s \end{matrix}

\right\)
Total number of ways = 276
Alternative
y can be from 2 to 9 so total number of ways =

\sum_{r = 2}^{9} (r^{2} - 1) = 276

Q. If

a! = a (a - 1) (a - 2) (a - 3) . . .1

, then find

X + Y + Z

, given that

X Y Z

is a three digit number such that

X Y Z = X! + Y! + Z!

Q. Even numbers are formed with three digits such that if

5

is one of tho digits then

7

is the next digit. The number of such numbers is (Repetition allowed.)

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Three digit numbers xyz are formed with digits 0,1,2,⋯9 such that x≤y≤z then number of such numbers is

x < y < z → 9C3 x = y < z → 9C2 x < y = z → 9C2 x = y = z → 9C1 _______ Total numbers =9C3+9C2+9C2+9C1=165

Three digit numbers $x y z$ are formed with digits $0, 1, 2, \dots 9$ such that $x \leq y \leq z$ then number of such numbers is

x < y < z $\to^{9} C_{3}$
x = y < z $\to^{9} C_{2}$
x < y = z $\to^{9} C_{2}$
x = y = z $\to^{9} C_{1}$
_______
Total numbers $=^{9} C_{3} +^{9} C_{2} +^{9} C_{2} +^{9} C_{1} = 165$