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 permutations for choosing and arranging 6 digits from 7 distinct digits

Number of ways = $^7P_6$

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

1. Determine the number of choices for each digit and apply the multiplication principle

There are $7$ available digits. Since repetition is allowed, each of the $6$ positions in the number can be filled in $7$ ways.
$\text{Number of ways} = 7^6$

2. Evaluate the expression

$7^6 = 117649$

1. Identify the possible frequency distributions for the digits making up the 6-digit number.

Since the number has 6 digits and every digit present must appear at least twice, the possible frequency distributions (partitions of 6 into parts $\ge 2$) are:

• Case 1: One digit appears 6 times (freq: $6$)
• Case 2: One digit appears 4 times, another appears 2 times (freq: $4, 2$)
• Case 3: Two distinct digits appear 3 times each (freq: $3, 3$)
• Case 4: Three distinct digits appear 2 times each (freq: $2, 2, 2$)

2. Calculate the number of valid arrangements for Cases 1 and 2.

Case 1 (6):
Number of ways to choose the digit = $\binom{7}{1} = 7$
Arrangements of this digit = $\frac{6!}{6!} = 1$
Number of ways = $7 \times 1 = 7$

Case 2 (4, 2):
Choose 1 digit to appear 4 times and 1 digit to appear 2 times: $\binom{7}{1} \times \binom{6}{1} = 42$ ways.
Arrangements = $\frac{6!}{4!2!} = 15$
Number of ways = $42 \times 15 = 630$

3. Calculate the number of valid arrangements for Cases 3 and 4.

Case 3 (3, 3):
Choose 2 digits: $\binom{7}{2} = 21$ ways (not $7 \times 6$, as the frequencies are identical, order of choosing does not matter).
Arrangements = $\frac{6!}{3!3!} = 20$
Number of ways = $21 \times 20 = 420$

Case 4 (2, 2, 2):
Choose 3 digits: $\binom{7}{3} = 35$ ways.
Arrangements = $\frac{6!}{2!2!2!} = 90$
Number of ways = $35 \times 90 = 3150$

4. Sum the possibilities to find the total number of 6-digit numbers.

$\text{Total valid numbers} = 7 + 630 + 420 + 3150 = 4207$