Question

An n digit  positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible, is

(a) 6

(b) 7

(c) 8

(d) 9​

Answer 1

Answer:

Using 2,5 and 7 with repetition each place of n digit number can be chosen in 3 ways.

Hence, total number of n−digit numbers =3×3×3...n times =3

n

.

According to given condition 3

n

≥900⇒3

n−2

≥100

∴n−2≥5⇒n≥7

Answer is 7.

Answer 2

$\large\underline{\sf{Solution-}}$

Given that,

An n digit  positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7.

Since, we have to formed n digit number, so it means there are n places to fill the given three digits 2, 5 and 7.

So, it means,

$\purple{\rm :\longmapsto\: {1}^{st} \: place \: can \: be \: filled \: in \: three \: ways}$

$\purple{\rm :\longmapsto\: {2}^{nd} \: place \: can \: be \: filled \: in \: three \: ways}$

$\purple{\rm :\longmapsto\: {3}^{rd} \: place \: can \: be \: filled \: in \: three \: ways}$

.

.

.

.

.

$\purple{\rm :\longmapsto\: {n}^{th} \: place \: can \: be \: filled \: in \: three \: ways}$

So, total number of ways in which n digit number can be formed with the help of the digits 2, 5 and 7 is

$\rm \:  =  \: \underbrace{3 \times 3 \times 3 \times 3 \times - - - \times 3} \\ \: \: \: \: \: \: \: \: n \: times$

$\rm \:  =  \: {3}^{n}$

Now, According to statement,

$\rm :\longmapsto\: {3}^{n} \geqslant 900$

$\rm :\longmapsto\: {3}^{n} \geqslant 9 \times 100$

$\rm :\longmapsto\: {3}^{n} \geqslant {3}^{2} \times 100$

$\rm :\longmapsto\: \dfrac{ {3}^{n} }{ {3}^{2} } \geqslant 100$

$\rm :\longmapsto\: {3}^{n - 2} \geqslant 100$

So, using hit and trial method,

$\purple{\rm :\longmapsto\: {3}^{4} = 81 \: \: \: and \: \: \: {3}^{5} = 243}$

So, it implies

$\rm :\longmapsto\:n - 2 \geqslant 5$

$\rm :\longmapsto\:n \geqslant 5 + 2$

$\rm\implies \:n \geqslant 7$

Hence,

• The smallest value of n is 7

So,

• Option (b) is correct.