How many six-digit numbers can be formed from the digits 1, 2, 3, 4, 5, 6 and 7 if

1. Calculate the number of ways to arrange 6 distinct digits chosen from 7 available digits

Number of ways = $^7P_6$
$= 7 \times 6 \times 5 \times 4 \times 3 \times 2$
$= 5040$

1. Apply the multiplication principle since each of the 6 digit positions can be filled by any of the 7 available digits

$\text{Number of ways} = 7 \times 7 \times 7 \times 7 \times 7 \times 7 = 7^6$
$7^6 = 117649$

1. Identify the possible frequency combinations of digits that sum to 6, where each digit used appears at least twice.

The 6-digit number must be formed by digits whose frequencies are $\ge 2$. We partition $6$ into sums of integers $\ge 2$:

• 3 distinct digits appearing 2 times each: $2+2+2=6$
• 2 distinct digits appearing 3 times each: $3+3=6$
• 2 distinct digits, one appearing 4 times and the other 2 times: $4+2=6$
• 1 digit appearing 6 times: $6$

2. Calculate the number of ways for the 2-2-2 frequency distribution.

For the 2-2-2 distribution:
Number of ways to choose 3 digits from 7: $\binom{7}{3} = 35$
Number of permutations of these 6 digits: $\frac{6!}{2!2!2!} = 90$
Number of numbers formed = $35 \times 90 = 3150$

3. Calculate the number of ways for the 3-3 and 4-2 frequency distributions.

For the 3-3 distribution:
Number of ways to choose 2 digits from 7: $\binom{7}{2} = 21$
Number of permutations of these 6 digits: $\frac{6!}{3!3!} = 20$
Number of numbers formed = $21 \times 20 = 420$

For the 4-2 distribution:
Number of ways to choose which digit appears 4 times and which appears 2 times: $7 \times 6 = 42$
Number of permutations of these 6 digits: $\frac{6!}{4!2!} = 15$
Number of numbers formed = $42 \times 15 = 630$

4. Calculate the number of ways for the 6 distribution and find the total sum.

For the 6 distribution:
Number of ways to choose 1 digit from 7: $\binom{7}{1} = 7$
Number of permutations: $\frac{6!}{6!} = 1$
Number of numbers formed = $7 \times 1 = 7$

Total possible 6-digit numbers = $3150 + 420 + 630 + 7 = 4207$