Does a mutual fund buy new shares when it issues new units for individuals?

Does a mutual fund buy new shares when it issues new units for individuals?
Does a mutual fund sell shares when we sold units of mutual fund?

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Yes.

Yes, that’s the whole point of mutual funds. But, you have to remember all funds hold cash, some more than other. Most funds typically have about 1-5% of the AUM as cash but sometimes some funds can go as high as 10%+, in some cases fresh inflows will be just cash.