Are private keys and mnemonic code one to one?

[hushan]

Do each private key corresponds to one and only one mnemonic phrase? and do each mnemonic phrase decode to the exact same private key always?

According to https://en.bitcoin.it/wiki/Private_key#Range_of_valid_ECDSA_private_keys, not all 2²⁵⁶ bits numbers are valid private keys, what about mnemonic phrases? Does all permutations of the words calculates to a valid private key? If not, does wallet software try multiple times to generate a valid phrase?

[ranochigo]

Do each private key corresponds to one and only one mnemonic phrase? and do each mnemonic phrase decode to the exact same private key always?

Mnemonic seeds are first converted to the master private key (seed) before being generated to a private key. Each private key can be generated from one and only one master private key since HMAC-SHA512 is applied to the master private before being generated and unless a collision is found, the hash will not be the same.

Mnemonic phrase will always decode to the same master private key and subsequently the same private key.

Does all permutations of the words calculates to a valid private key? If not, does wallet software try multiple times to generate a valid phrase?

Yes, according to BIP39 standards. You can select the words from the wordlist yourself and you can generate a seed from using any mnemorics. However, depending on your client, you might need to have the correct checksum.

[Heisenberg_Hunter]

According to https://en.bitcoin.it/wiki/Private_key#Range_of_valid_ECDSA_private_keys, not all 2²⁵⁶ bits numbers are valid private keys, what about mnemonic phrases?

According to BIP39 standards, we can use around 2048 words to generate a mnemonic key. For us to generate a 12 word key pair the combinations could range between 2048¹² which could be simplified to 2¹³² and the mnemonic would eventually have 132 bits of security. But due to some randomness, the security could be around 128 bits which is equal to a private key security.

But generating a word yourself from the 2048 lists is pretty dangerous when security is concerned. Allow your wallet to work for you. For a complete understanding of BIP39 conventions you may read this