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4 people will receive both a t-shirt and a keychain.  

Step-by-step explanation:

To determine how many people will receive both a t-shirt and a keychain,

First we will determine those that will receive a t-shirt and as well determine those that will receive a keychain.

For those that will a receive a t-shirt

Since every 8th person receives a t-shirt, then

The 8th, 16th, 24th, 32nd, 40th, 48th, 56th, 64th, 72nd, 80th, 88th, 96th, 104th, 112th. 120th, 128th, 136th, 144th, 152nd, and 160th persons will receive a t-shirt.

For those that will receive a keychain

Since every 10th person receives a keychain, then

The 10th, 20th, 30th, 40th, 50th, 60th, 70th, 80th, 90th, 100th, 110th, 120th, 130th, 140th, 150th, and 160th persons will receive a keychain.

From above, those that will receive both a t-shirt and a keychain are the 40th, 80th, 120th, and 160th persons.

Hence, 4 people will receive both a t-shirt and a keychain.