An-eight-digit-number-is-formed-from-1-2-3-4-such-that-product-of-all-digits-is-always-3072-the-total-number-of-ways-is-23-8-C-k-where-the-value-of-k-is-

Question Number 21930 by Tinkutara last updated on 07/Oct/17

An eight digit number is formed from

1, 2, 3, 4 such that product of all digits

is always 3072, the total number of

ways is (23 .^{8} C_{k}), where the value of k

is

Commented by mrW1 last updated on 07/Oct/17

k = 3 or 5 ?

Commented by mrW1 last updated on 07/Oct/17

for k = 1 or 7 : 23 \times C_{k}^{8} = 23 \times 8 = 184

for k = 2 or 6 : 23 \times C_{k}^{8} = 23 \times 28 = 644

for k = 3 or 5 : 23 \times C_{k}^{8} = 23 \times 64 = 1288

for k = 4 : 23 \times C_{k}^{8} = 23 \times 70 = 1610

3072 = 1^{2} \times 3^{1} \times 4^{5} \Rightarrow case 1

3072 = 1^{1} \times 2^{2} \times 3^{1} \times 4^{4} \Rightarrow case 2

3072 = 2^{4} \times 3^{1} \times 4^{3} \Rightarrow case 3

case 1 : 2 digits 1, 1 digit 3, 5 digits 4

\Rightarrow C_{2}^{8} \times C_{1}^{6} = 28 \times 6 = 168

case 2 : 1 digit 1, 2 digits 2, 1 digit 3, 4 digits 4

\Rightarrow C_{1}^{8} \times C_{2}^{7} \times C_{1}^{5} = 8 \times 21 \times 5 = 840

case 3 : 4 digits 2, 1 digit 3, 3 digits 4

\Rightarrow C_{4}^{8} \times C_{1}^{4} = 70 \times 4 = 280

\Rightarrow total ways :

168 + 840 + 280 = 1288

\Rightarrow k = 3 or 5

Commented by Tinkutara last updated on 07/Oct/17

Thank you very much Sir!