Answer: 2399760
Step-by-step explanation:
The concept we use here is Partial derangement.
It says that for m things , the number of ways to arrange them such that k things are not in their fixed position is given by :-
[tex]m!-^kC_1(m-1)!+^kC_2(m-2)!-^kC_3(m-3)!+........[/tex]
Given digits : 0,1,2,3,4,5,6,7,8,9
Prime numbers = 2,3,5,7
Now by Partial derangement the number of ways to arrange 10 numbers such that none of 4 prime numbers is in its original position will be :_
[tex]10!-^4C_1(9)!+^4C_2(8)!-^4C_3(7)!+^4C_4(6)!\\\\=3628800-(4)(362880)+\dfrac{4!}{2!2!}(40320)-(4)(5040)+(1)(720)\\\\=3628800-1451520+241920-20160+720\\\\=2399760[/tex]
Hence, the number of ways can the digits 0,1,2,3,4,5,6,7,8,9 be arranged so that no prime number is in its original position = 2399760
