Question

Using multinominal coefficients:

How many DNA strings of length 13 contains 4 As, 2 Cs, 3 Gs and 4
Ts and are palendromes?

Answer 1

The number of DNA strings of length 13 that contain 4 As, 2 Cs, 3 Gs, and 4 Ts and are palindromes is 27,405.

Step-by-step explanation:

To find the number of DNA strings of length 13 that contain 4 As, 2 Cs, 3 Gs, and 4 Ts and are palindromes, we can use multinomial coefficients.

A palindromic DNA string of length 13 will have the following structure: A _ _ _ _ _ T _ _ _ _ _ A, where the blanks can be filled with the other nucleotides.

Therefore, the number of palindromic DNA strings can be calculated as the multinomial coefficient of the number of As, Cs, Gs, and Ts: C(4, 2, 3, 4) = 13! / (4! * 2! * 3! * 4!) = 27,405.

Answer 2

To find the number of DNA strings of length 13 that contain 4 As, 2 Cs, 3 Gs, and 4 Ts and are palindromes, we can use multinomial coefficients.

Step-by-step explanation:

To find the number of DNA strings of length 13 that contain 4 As, 2 Cs, 3 Gs, and 4 Ts and are palindromes, we can use multinomial coefficients.

A palindrome is a string that reads the same forwards and backwards.

Since there are 13 total positions in the string, and we know the counts of each nucleotide, we can assign the positions in a palindrome in the following way: The first half of the positions will contain the counts of As, Cs, and Gs, and the second half will contain the counts of Ts in reverse order.

For example, ACGTTAAGTTGCA is a palindrome that satisfies the given conditions.

The number of ways to arrange the As, Cs, and Gs in the first half of the positions is given by the multinomial coefficient (13 choose 4, 2, 3).

Therefore, the number of DNA strings of length 13 that contain 4 As, 2 Cs, 3 Gs, and 4 Ts and are palindromes is (13 choose 4, 2, 3).